[{"data":1,"prerenderedAt":9953},["ShallowReactive",2],{"learn-\u002Flearn\u002Ftools\u002Ftienstra-resection":3,"learn-\u002Flearn\u002Ftools\u002Ftienstra-resection-surround":9942},{"id":4,"title":5,"body":6,"category":9932,"description":9933,"extension":9934,"meta":9935,"navigation":9936,"order":9896,"path":9937,"seo":9938,"sitemap":9939,"stem":9940,"__hash__":9941},"learn\u002Flearn\u002Ftools\u002Ftienstra-resection.md","Tienstra Resection",{"type":7,"value":8,"toc":9889},"minimark",[9,14,18,80,84,87,92,95,99,227,417,421,486,724,1349,1944,2045,2049,2052,2056,2250,2318,2727,2730,2868,2872,2954,2957,3218,3222,3480,3491,3495,3499,3510,3514,3643,3857,3860,4145,4337,4341,4344,4348,4352,4473,4651,4872,4876,5078,5297,5510,5725,5757,6918,6922,6926,6977,7009,7396,7499,7546,8354,8389,8393,8550,8722,9085,9151,9155,9341,9347,9351,9355,9545,9549,9680,9684,9688,9781,9785,9873,9877,9880,9884],[10,11,13],"h2",{"id":12},"planimetric-positioning-via-the-tienstra-three-point-resection","Planimetric Positioning via the Tienstra Three-Point Resection",[15,16,17],"p",{},"In engineering geodesy, establishing high-precision horizontal control is a prerequisite for total station setups and subsequent stakeout or monitoring. GNSS is often blocked in urban canyons, open pits, and forest canopy, forcing crews to rely on classical optical methods. When distances to control cannot be measured — due to hazards, obstacles, or coordinate-only networks — the surveyor must solve position from angles alone.",[15,19,20,21,25,26,73,74,79],{},"The three-point angular resection establishes a ",[22,23,24],"strong",{},"free station"," at an unknown point ",[27,28,31,53],"span",{"className":29},[30],"katex",[27,32,35],{"className":33},[34],"katex-mathml",[36,37,39],"math",{"xmlns":38},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[40,41,42,49],"semantics",{},[43,44,45],"mrow",{},[46,47,48],"mi",{},"P",[50,51,48],"annotation",{"encoding":52},"application\u002Fx-tex",[27,54,58],{"className":55,"ariaHidden":57},[56],"katex-html","true",[27,59,62,67],{"className":60},[61],"base",[27,63],{"className":64,"style":66},[65],"strut","height:0.6833em;",[27,68,48],{"className":69,"style":72},[70,71],"mord","mathnormal","margin-right:0.1389em;"," by measuring horizontal angles to three visible, coordinated benchmarks. The ",[75,76,78],"a",{"href":77},"\u002Ftools\u002Ftienstra-resection","Tienstra resection calculator"," implements the cotangent barycentric formulation for rapid coordinate resolution from field inputs. This article covers the geometry, danger circle, field practice, computation, and verification.",[10,81,83],{"id":82},"defining-the-resection-problem-resection-vs-intersection","Defining the Resection Problem: Resection vs. Intersection",[15,85,86],{},"Planimetric positioning divides into two primary configurations:",[88,89,91],"h3",{"id":90},"intersection","Intersection",[15,93,94],{},"The unknown point is an inaccessible target (peak, spire, remote monument). Instruments occupy at least two known stations; bearings from each station intersect at the target. Accurate, but requires multiple setups and access to control.",[88,96,98],{"id":97},"resection","Resection",[15,100,101,102,105,106,134,135,164,165,195,196,226],{},"The surveyor occupies the ",[22,103,104],{},"unknown"," point ",[27,107,109,122],{"className":108},[30],[27,110,112],{"className":111},[34],[36,113,114],{"xmlns":38},[40,115,116,120],{},[43,117,118],{},[46,119,48],{},[50,121,48],{"encoding":52},[27,123,125],{"className":124,"ariaHidden":57},[56],[27,126,128,131],{"className":127},[61],[27,129],{"className":130,"style":66},[65],[27,132,48],{"className":133,"style":72},[70,71]," and observes horizontal angles between three known stations ",[27,136,138,152],{"className":137},[30],[27,139,141],{"className":140},[34],[36,142,143],{"xmlns":38},[40,144,145,150],{},[43,146,147],{},[46,148,149],{},"A",[50,151,149],{"encoding":52},[27,153,155],{"className":154,"ariaHidden":57},[56],[27,156,158,161],{"className":157},[61],[27,159],{"className":160,"style":66},[65],[27,162,149],{"className":163},[70,71],", ",[27,166,168,182],{"className":167},[30],[27,169,171],{"className":170},[34],[36,172,173],{"xmlns":38},[40,174,175,180],{},[43,176,177],{},[46,178,179],{},"B",[50,181,179],{"encoding":52},[27,183,185],{"className":184,"ariaHidden":57},[56],[27,186,188,191],{"className":187},[61],[27,189],{"className":190,"style":66},[65],[27,192,179],{"className":193,"style":194},[70,71],"margin-right:0.0502em;",", and ",[27,197,199,213],{"className":198},[30],[27,200,202],{"className":201},[34],[36,203,204],{"xmlns":38},[40,205,206,211],{},[43,207,208],{},[46,209,210],{},"C",[50,212,210],{"encoding":52},[27,214,216],{"className":215,"ariaHidden":57},[56],[27,217,219,222],{"className":218},[61],[27,220],{"className":221,"style":66},[65],[27,223,210],{"className":224,"style":225},[70,71],"margin-right:0.0715em;",". Only one setup is required; control points need not be occupied — only visible. Ideal for construction site free-stationing referencing distant stable benchmarks.",[228,229,230,246],"table",{},[231,232,233],"thead",{},[234,235,236,240,243],"tr",{},[237,238,239],"th",{},"Parameter",[237,241,242],{},"Angular intersection",[237,244,245],{},"Angular resection",[247,248,249,289,356,395,406],"tbody",{},[234,250,251,255,258],{},[252,253,254],"td",{},"Occupied station(s)",[252,256,257],{},"Multiple known points",[252,259,260,261],{},"Single unknown point ",[27,262,264,277],{"className":263},[30],[27,265,267],{"className":266},[34],[36,268,269],{"xmlns":38},[40,270,271,275],{},[43,272,273],{},[46,274,48],{},[50,276,48],{"encoding":52},[27,278,280],{"className":279,"ariaHidden":57},[56],[27,281,283,286],{"className":282},[61],[27,284],{"className":285,"style":66},[65],[27,287,48],{"className":288,"style":72},[70,71],[234,290,291,294,325],{},[252,292,293],{},"Observation type",[252,295,296,297],{},"Bearings from known points to ",[27,298,300,313],{"className":299},[30],[27,301,303],{"className":302},[34],[36,304,305],{"xmlns":38},[40,306,307,311],{},[43,308,309],{},[46,310,48],{},[50,312,48],{"encoding":52},[27,314,316],{"className":315,"ariaHidden":57},[56],[27,317,319,322],{"className":318},[61],[27,320],{"className":321,"style":66},[65],[27,323,48],{"className":324,"style":72},[70,71],[252,326,327,328],{},"Angles between known points at ",[27,329,331,344],{"className":330},[30],[27,332,334],{"className":333},[34],[36,335,336],{"xmlns":38},[40,337,338,342],{},[43,339,340],{},[46,341,48],{},[50,343,48],{"encoding":52},[27,345,347],{"className":346,"ariaHidden":57},[56],[27,348,350,353],{"className":349},[61],[27,351],{"className":352,"style":66},[65],[27,354,48],{"className":355,"style":72},[70,71],[234,357,358,361,364],{},[252,359,360],{},"Minimum requirements",[252,362,363],{},"Two known points, intervisible baseline",[252,365,366,367],{},"Three non-collinear known points visible from ",[27,368,370,383],{"className":369},[30],[27,371,373],{"className":372},[34],[36,374,375],{"xmlns":38},[40,376,377,381],{},[43,378,379],{},[46,380,48],{},[50,382,48],{"encoding":52},[27,384,386],{"className":385,"ariaHidden":57},[56],[27,387,389,392],{"className":388},[61],[27,390],{"className":391,"style":66},[65],[27,393,48],{"className":394,"style":72},[70,71],[234,396,397,400,403],{},[252,398,399],{},"Operational advantage",[252,401,402],{},"Inaccessible targets",[252,404,405],{},"Rapid free-stationing",[234,407,408,411,414],{},[252,409,410],{},"Logistical effort",[252,412,413],{},"High (multiple crews\u002Fsetups)",[252,415,416],{},"Low (single setup)",[10,418,420],{"id":419},"the-tienstra-three-point-method","The Tienstra Three-Point Method",[15,422,423,424,427,428,456,457,485],{},"Classical geometric methods (Cassini, Collins) use auxiliary constructions and are error-prone in manual calculation. The ",[22,425,426],{},"Tienstra formula"," — popularized by J.M. Tienstra (1895–1951) at Delft — solves ",[27,429,431,444],{"className":430},[30],[27,432,434],{"className":433},[34],[36,435,436],{"xmlns":38},[40,437,438,442],{},[43,439,440],{},[46,441,48],{},[50,443,48],{"encoding":52},[27,445,447],{"className":446,"ariaHidden":57},[56],[27,448,450,453],{"className":449},[61],[27,451],{"className":452,"style":66},[65],[27,454,48],{"className":455,"style":72},[70,71]," directly with barycentric weights from cotangents of control-triangle interior angles and the observed angles at ",[27,458,460,473],{"className":459},[30],[27,461,463],{"className":462},[34],[36,464,465],{"xmlns":38},[40,466,467,471],{},[43,468,469],{},[46,470,48],{},[50,472,48],{"encoding":52},[27,474,476],{"className":475,"ariaHidden":57},[56],[27,477,479,482],{"className":478},[61],[27,480],{"className":481,"style":66},[65],[27,483,48],{"className":484,"style":72},[70,71],".",[15,487,488,489,164,581,164,652,723],{},"Once weights ",[27,490,492,514],{"className":491},[30],[27,493,495],{"className":494},[34],[36,496,497],{"xmlns":38},[40,498,499,511],{},[43,500,501],{},[502,503,504,507],"msub",{},[46,505,506],{},"f",[508,509,510],"mn",{},"1",[50,512,513],{"encoding":52},"f_1",[27,515,517],{"className":516,"ariaHidden":57},[56],[27,518,520,524],{"className":519},[61],[27,521],{"className":522,"style":523},[65],"height:0.8889em;vertical-align:-0.1944em;",[27,525,527,531],{"className":526},[70],[27,528,506],{"className":529,"style":530},[70,71],"margin-right:0.1076em;",[27,532,535],{"className":533},[534],"msupsub",[27,536,540,572],{"className":537},[538,539],"vlist-t","vlist-t2",[27,541,544,567],{"className":542},[543],"vlist-r",[27,545,549],{"className":546,"style":548},[547],"vlist","height:0.3011em;",[27,550,552,557],{"style":551},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[27,553],{"className":554,"style":556},[555],"pstrut","height:2.7em;",[27,558,564],{"className":559},[560,561,562,563],"sizing","reset-size6","size3","mtight",[27,565,510],{"className":566},[70,563],[27,568,571],{"className":569},[570],"vlist-s","​",[27,573,575],{"className":574},[543],[27,576,579],{"className":577,"style":578},[547],"height:0.15em;",[27,580],{},[27,582,584,603],{"className":583},[30],[27,585,587],{"className":586},[34],[36,588,589],{"xmlns":38},[40,590,591,600],{},[43,592,593],{},[502,594,595,597],{},[46,596,506],{},[508,598,599],{},"2",[50,601,602],{"encoding":52},"f_2",[27,604,606],{"className":605,"ariaHidden":57},[56],[27,607,609,612],{"className":608},[61],[27,610],{"className":611,"style":523},[65],[27,613,615,618],{"className":614},[70],[27,616,506],{"className":617,"style":530},[70,71],[27,619,621],{"className":620},[534],[27,622,624,644],{"className":623},[538,539],[27,625,627,641],{"className":626},[543],[27,628,630],{"className":629,"style":548},[547],[27,631,632,635],{"style":551},[27,633],{"className":634,"style":556},[555],[27,636,638],{"className":637},[560,561,562,563],[27,639,599],{"className":640},[70,563],[27,642,571],{"className":643},[570],[27,645,647],{"className":646},[543],[27,648,650],{"className":649,"style":578},[547],[27,651],{},[27,653,655,674],{"className":654},[30],[27,656,658],{"className":657},[34],[36,659,660],{"xmlns":38},[40,661,662,671],{},[43,663,664],{},[502,665,666,668],{},[46,667,506],{},[508,669,670],{},"3",[50,672,673],{"encoding":52},"f_3",[27,675,677],{"className":676,"ariaHidden":57},[56],[27,678,680,683],{"className":679},[61],[27,681],{"className":682,"style":523},[65],[27,684,686,689],{"className":685},[70],[27,687,506],{"className":688,"style":530},[70,71],[27,690,692],{"className":691},[534],[27,693,695,715],{"className":694},[538,539],[27,696,698,712],{"className":697},[543],[27,699,701],{"className":700,"style":548},[547],[27,702,703,706],{"style":551},[27,704],{"className":705,"style":556},[555],[27,707,709],{"className":708},[560,561,562,563],[27,710,670],{"className":711},[70,563],[27,713,571],{"className":714},[570],[27,716,718],{"className":717},[543],[27,719,721],{"className":720,"style":578},[547],[27,722],{}," 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",[27,1948,1950,1963],{"className":1949},[30],[27,1951,1953],{"className":1952},[34],[36,1954,1955],{"xmlns":38},[40,1956,1957,1961],{},[43,1958,1959],{},[46,1960,48],{},[50,1962,48],{"encoding":52},[27,1964,1966],{"className":1965,"ariaHidden":57},[56],[27,1967,1969,1972],{"className":1968},[61],[27,1970],{"className":1971,"style":66},[65],[27,1973,48],{"className":1974,"style":72},[70,71]," is the weighted balance of triangle ",[27,1977,1979,1997],{"className":1978},[30],[27,1980,1982],{"className":1981},[34],[36,1983,1984],{"xmlns":38},[40,1985,1986,1994],{},[43,1987,1988,1990,1992],{},[46,1989,149],{},[46,1991,179],{},[46,1993,210],{},[50,1995,1996],{"encoding":52},"ABC",[27,1998,2000],{"className":1999,"ariaHidden":57},[56],[27,2001,2003,2006,2009,2012],{"className":2002},[61],[27,2004],{"className":2005,"style":66},[65],[27,2007,149],{"className":2008},[70,71],[27,2010,179],{"className":2011,"style":194},[70,71],[27,2013,210],{"className":2014,"style":225},[70,71]," — the formula handles ",[27,2017,2019,2032],{"className":2018},[30],[27,2020,2022],{"className":2021},[34],[36,2023,2024],{"xmlns":38},[40,2025,2026,2030],{},[43,2027,2028],{},[46,2029,48],{},[50,2031,48],{"encoding":52},[27,2033,2035],{"className":2034,"ariaHidden":57},[56],[27,2036,2038,2041],{"className":2037},[61],[27,2039],{"className":2040,"style":66},[65],[27,2042,48],{"className":2043,"style":72},[70,71]," inside or outside the reference triangle.",[10,2046,2048],{"id":2047},"control-geometry-and-the-danger-circle","Control Geometry and the Danger Circle",[15,2050,2051],{},"Resection precision depends on the spatial arrangement of control and the unknown station. Poor geometry magnifies angular errors or causes failure.",[88,2053,2055],{"id":2054},"point-ordering-and-angle-closure","Point ordering and angle closure",[15,2057,2058,2059,164,2087,164,2115,2143,2144,2147,2148,2186,2187,2217,2218,2249],{},"Sequence control points ",[27,2060,2062,2075],{"className":2061},[30],[27,2063,2065],{"className":2064},[34],[36,2066,2067],{"xmlns":38},[40,2068,2069,2073],{},[43,2070,2071],{},[46,2072,149],{},[50,2074,149],{"encoding":52},[27,2076,2078],{"className":2077,"ariaHidden":57},[56],[27,2079,2081,2084],{"className":2080},[61],[27,2082],{"className":2083,"style":66},[65],[27,2085,149],{"className":2086},[70,71],[27,2088,2090,2103],{"className":2089},[30],[27,2091,2093],{"className":2092},[34],[36,2094,2095],{"xmlns":38},[40,2096,2097,2101],{},[43,2098,2099],{},[46,2100,179],{},[50,2102,179],{"encoding":52},[27,2104,2106],{"className":2105,"ariaHidden":57},[56],[27,2107,2109,2112],{"className":2108},[61],[27,2110],{"className":2111,"style":66},[65],[27,2113,179],{"className":2114,"style":194},[70,71],[27,2116,2118,2131],{"className":2117},[30],[27,2119,2121],{"className":2120},[34],[36,2122,2123],{"xmlns":38},[40,2124,2125,2129],{},[43,2126,2127],{},[46,2128,210],{},[50,2130,210],{"encoding":52},[27,2132,2134],{"className":2133,"ariaHidden":57},[56],[27,2135,2137,2140],{"className":2136},[61],[27,2138],{"className":2139,"style":66},[65],[27,2141,210],{"className":2142,"style":225},[70,71]," ",[22,2145,2146],{},"counterclockwise"," around triangle ",[27,2149,2151,2168],{"className":2150},[30],[27,2152,2154],{"className":2153},[34],[36,2155,2156],{"xmlns":38},[40,2157,2158,2166],{},[43,2159,2160,2162,2164],{},[46,2161,149],{},[46,2163,179],{},[46,2165,210],{},[50,2167,1996],{"encoding":52},[27,2169,2171],{"className":2170,"ariaHidden":57},[56],[27,2172,2174,2177,2180,2183],{"className":2173},[61],[27,2175],{"className":2176,"style":66},[65],[27,2178,149],{"className":2179},[70,71],[27,2181,179],{"className":2182,"style":194},[70,71],[27,2184,210],{"className":2185,"style":225},[70,71]," (right-handed grid: easting = ",[27,2188,2190,2204],{"className":2189},[30],[27,2191,2193],{"className":2192},[34],[36,2194,2195],{"xmlns":38},[40,2196,2197,2202],{},[43,2198,2199],{},[46,2200,2201],{},"x",[50,2203,2201],{"encoding":52},[27,2205,2207],{"className":2206,"ariaHidden":57},[56],[27,2208,2210,2214],{"className":2209},[61],[27,2211],{"className":2212,"style":2213},[65],"height:0.4306em;",[27,2215,2201],{"className":2216},[70,71],", northing = ",[27,2219,2221,2235],{"className":2220},[30],[27,2222,2224],{"className":2223},[34],[36,2225,2226],{"xmlns":38},[40,2227,2228,2233],{},[43,2229,2230],{},[46,2231,2232],{},"y",[50,2234,2232],{"encoding":52},[27,2236,2238],{"className":2237,"ariaHidden":57},[56],[27,2239,2241,2245],{"className":2240},[61],[27,2242],{"className":2243,"style":2244},[65],"height:0.625em;vertical-align:-0.1944em;",[27,2246,2232],{"className":2247,"style":2248},[70,71],"margin-right:0.0359em;",").",[15,2251,2252,2253,2281,2282,2317],{},"At unknown station ",[27,2254,2256,2269],{"className":2255},[30],[27,2257,2259],{"className":2258},[34],[36,2260,2261],{"xmlns":38},[40,2262,2263,2267],{},[43,2264,2265],{},[46,2266,48],{},[50,2268,48],{"encoding":52},[27,2270,2272],{"className":2271,"ariaHidden":57},[56],[27,2273,2275,2278],{"className":2274},[61],[27,2276],{"className":2277,"style":66},[65],[27,2279,48],{"className":2280,"style":72},[70,71],", measure horizon angles clockwise around the full ",[27,2283,2285,2304],{"className":2284},[30],[27,2286,2288],{"className":2287},[34],[36,2289,2290],{"xmlns":38},[40,2291,2292,2301],{},[43,2293,2294,2297],{},[508,2295,2296],{},"360",[46,2298,2300],{"mathvariant":2299},"normal","°",[50,2302,2303],{"encoding":52},"360°",[27,2305,2307],{"className":2306,"ariaHidden":57},[56],[27,2308,2310,2314],{"className":2309},[61],[27,2311],{"className":2312,"style":2313},[65],"height:0.6944em;",[27,2315,2303],{"className":2316},[70],":",[2319,2320,2321,2461,2594],"ul",{},[2322,2323,2324,2355,2356,2402,2403,2431,2432,2460],"li",{},[27,2325,2327,2342],{"className":2326},[30],[27,2328,2330],{"className":2329},[34],[36,2331,2332],{"xmlns":38},[40,2333,2334,2339],{},[43,2335,2336],{},[46,2337,2338],{},"α",[50,2340,2341],{"encoding":52},"\\alpha",[27,2343,2345],{"className":2344,"ariaHidden":57},[56],[27,2346,2348,2351],{"className":2347},[61],[27,2349],{"className":2350,"style":2213},[65],[27,2352,2338],{"className":2353,"style":2354},[70,71],"margin-right:0.0037em;"," = ",[27,2357,2359,2380],{"className":2358},[30],[27,2360,2362],{"className":2361},[34],[36,2363,2364],{"xmlns":38},[40,2365,2366,2377],{},[43,2367,2368,2371,2373,2375],{},[46,2369,2370],{"mathvariant":2299},"∠",[46,2372,149],{},[46,2374,48],{},[46,2376,179],{},[50,2378,2379],{"encoding":52},"\\angle APB",[27,2381,2383],{"className":2382,"ariaHidden":57},[56],[27,2384,2386,2390,2393,2396,2399],{"className":2385},[61],[27,2387],{"className":2388,"style":2389},[65],"height:0.6922em;",[27,2391,2370],{"className":2392},[70],[27,2394,149],{"className":2395},[70,71],[27,2397,48],{"className":2398,"style":72},[70,71],[27,2400,179],{"className":2401,"style":194},[70,71]," (from ",[27,2404,2406,2419],{"className":2405},[30],[27,2407,2409],{"className":2408},[34],[36,2410,2411],{"xmlns":38},[40,2412,2413,2417],{},[43,2414,2415],{},[46,2416,149],{},[50,2418,149],{"encoding":52},[27,2420,2422],{"className":2421,"ariaHidden":57},[56],[27,2423,2425,2428],{"className":2424},[61],[27,2426],{"className":2427,"style":66},[65],[27,2429,149],{"className":2430},[70,71]," to ",[27,2433,2435,2448],{"className":2434},[30],[27,2436,2438],{"className":2437},[34],[36,2439,2440],{"xmlns":38},[40,2441,2442,2446],{},[43,2443,2444],{},[46,2445,179],{},[50,2447,179],{"encoding":52},[27,2449,2451],{"className":2450,"ariaHidden":57},[56],[27,2452,2454,2457],{"className":2453},[61],[27,2455],{"className":2456,"style":66},[65],[27,2458,179],{"className":2459,"style":194},[70,71],")",[2322,2462,2463,2355,2494,2402,2538,2431,2566,2460],{},[27,2464,2466,2481],{"className":2465},[30],[27,2467,2469],{"className":2468},[34],[36,2470,2471],{"xmlns":38},[40,2472,2473,2478],{},[43,2474,2475],{},[46,2476,2477],{},"β",[50,2479,2480],{"encoding":52},"\\beta",[27,2482,2484],{"className":2483,"ariaHidden":57},[56],[27,2485,2487,2490],{"className":2486},[61],[27,2488],{"className":2489,"style":523},[65],[27,2491,2477],{"className":2492,"style":2493},[70,71],"margin-right:0.0528em;",[27,2495,2497,2517],{"className":2496},[30],[27,2498,2500],{"className":2499},[34],[36,2501,2502],{"xmlns":38},[40,2503,2504,2514],{},[43,2505,2506,2508,2510,2512],{},[46,2507,2370],{"mathvariant":2299},[46,2509,179],{},[46,2511,48],{},[46,2513,210],{},[50,2515,2516],{"encoding":52},"\\angle BPC",[27,2518,2520],{"className":2519,"ariaHidden":57},[56],[27,2521,2523,2526,2529,2532,2535],{"className":2522},[61],[27,2524],{"className":2525,"style":2389},[65],[27,2527,2370],{"className":2528},[70],[27,2530,179],{"className":2531,"style":194},[70,71],[27,2533,48],{"className":2534,"style":72},[70,71],[27,2536,210],{"className":2537,"style":225},[70,71],[27,2539,2541,2554],{"className":2540},[30],[27,2542,2544],{"className":2543},[34],[36,2545,2546],{"xmlns":38},[40,2547,2548,2552],{},[43,2549,2550],{},[46,2551,179],{},[50,2553,179],{"encoding":52},[27,2555,2557],{"className":2556,"ariaHidden":57},[56],[27,2558,2560,2563],{"className":2559},[61],[27,2561],{"className":2562,"style":66},[65],[27,2564,179],{"className":2565,"style":194},[70,71],[27,2567,2569,2582],{"className":2568},[30],[27,2570,2572],{"className":2571},[34],[36,2573,2574],{"xmlns":38},[40,2575,2576,2580],{},[43,2577,2578],{},[46,2579,210],{},[50,2581,210],{"encoding":52},[27,2583,2585],{"className":2584,"ariaHidden":57},[56],[27,2586,2588,2591],{"className":2587},[61],[27,2589],{"className":2590,"style":66},[65],[27,2592,210],{"className":2593,"style":225},[70,71],[2322,2595,2596,2355,2627,2402,2671,2431,2699,2460],{},[27,2597,2599,2614],{"className":2598},[30],[27,2600,2602],{"className":2601},[34],[36,2603,2604],{"xmlns":38},[40,2605,2606,2611],{},[43,2607,2608],{},[46,2609,2610],{},"γ",[50,2612,2613],{"encoding":52},"\\gamma",[27,2615,2617],{"className":2616,"ariaHidden":57},[56],[27,2618,2620,2623],{"className":2619},[61],[27,2621],{"className":2622,"style":2244},[65],[27,2624,2610],{"className":2625,"style":2626},[70,71],"margin-right:0.0556em;",[27,2628,2630,2650],{"className":2629},[30],[27,2631,2633],{"className":2632},[34],[36,2634,2635],{"xmlns":38},[40,2636,2637,2647],{},[43,2638,2639,2641,2643,2645],{},[46,2640,2370],{"mathvariant":2299},[46,2642,210],{},[46,2644,48],{},[46,2646,149],{},[50,2648,2649],{"encoding":52},"\\angle CPA",[27,2651,2653],{"className":2652,"ariaHidden":57},[56],[27,2654,2656,2659,2662,2665,2668],{"className":2655},[61],[27,2657],{"className":2658,"style":2389},[65],[27,2660,2370],{"className":2661},[70],[27,2663,210],{"className":2664,"style":225},[70,71],[27,2666,48],{"className":2667,"style":72},[70,71],[27,2669,149],{"className":2670},[70,71],[27,2672,2674,2687],{"className":2673},[30],[27,2675,2677],{"className":2676},[34],[36,2678,2679],{"xmlns":38},[40,2680,2681,2685],{},[43,2682,2683],{},[46,2684,210],{},[50,2686,210],{"encoding":52},[27,2688,2690],{"className":2689,"ariaHidden":57},[56],[27,2691,2693,2696],{"className":2692},[61],[27,2694],{"className":2695,"style":66},[65],[27,2697,210],{"className":2698,"style":225},[70,71],[27,2700,2702,2715],{"className":2701},[30],[27,2703,2705],{"className":2704},[34],[36,2706,2707],{"xmlns":38},[40,2708,2709,2713],{},[43,2710,2711],{},[46,2712,149],{},[50,2714,149],{"encoding":52},[27,2716,2718],{"className":2717,"ariaHidden":57},[56],[27,2719,2721,2724],{"className":2720},[61],[27,2722],{"className":2723,"style":66},[65],[27,2725,149],{"className":2726},[70,71],[15,2728,2729],{},"Closure requirement:",[15,2731,2732],{},[27,2733,2735,2767],{"className":2734},[30],[27,2736,2738],{"className":2737},[34],[36,2739,2740],{"xmlns":38},[40,2741,2742,2764],{},[43,2743,2744,2746,2748,2750,2752,2754,2756],{},[46,2745,2338],{},[746,2747,768],{},[46,2749,2477],{},[746,2751,768],{},[46,2753,2610],{},[746,2755,748],{},[2757,2758,2759,2761],"msup",{},[508,2760,2296],{},[746,2762,2763],{},"∘",[50,2765,2766],{"encoding":52},"\\alpha + \\beta + \\gamma = 360^\\circ",[27,2768,2770,2790,2808,2826],{"className":2769,"ariaHidden":57},[56],[27,2771,2773,2777,2780,2784,2787],{"className":2772},[61],[27,2774],{"className":2775,"style":2776},[65],"height:0.6667em;vertical-align:-0.0833em;",[27,2778,2338],{"className":2779,"style":2354},[70,71],[27,2781],{"className":2782,"style":2783},[878],"margin-right:0.2222em;",[27,2785,768],{"className":2786},[977],[27,2788],{"className":2789,"style":2783},[878],[27,2791,2793,2796,2799,2802,2805],{"className":2792},[61],[27,2794],{"className":2795,"style":523},[65],[27,2797,2477],{"className":2798,"style":2493},[70,71],[27,2800],{"className":2801,"style":2783},[878],[27,2803,768],{"className":2804},[977],[27,2806],{"className":2807,"style":2783},[878],[27,2809,2811,2814,2817,2820,2823],{"className":2810},[61],[27,2812],{"className":2813,"style":2244},[65],[27,2815,2610],{"className":2816,"style":2626},[70,71],[27,2818],{"className":2819,"style":879},[878],[27,2821,748],{"className":2822},[883],[27,2824],{"className":2825,"style":879},[878],[27,2827,2829,2833,2837],{"className":2828},[61],[27,2830],{"className":2831,"style":2832},[65],"height:0.6741em;",[27,2834,2836],{"className":2835},[70],"36",[27,2838,2840,2844],{"className":2839},[70],[27,2841,2843],{"className":2842},[70],"0",[27,2845,2847],{"className":2846},[534],[27,2848,2850],{"className":2849},[538],[27,2851,2853],{"className":2852},[543],[27,2854,2856],{"className":2855,"style":2832},[547],[27,2857,2859,2862],{"style":2858},"top:-3.063em;margin-right:0.05em;",[27,2860],{"className":2861,"style":556},[555],[27,2863,2865],{"className":2864},[560,561,562,563],[27,2866,2763],{"className":2867},[977,563],[88,2869,2871],{"id":2870},"the-danger-circle","The danger circle",[15,2873,2874,2875,2878,2879,2924,2925,2953],{},"The critical singularity is the ",[22,2876,2877],{},"danger circle"," (circumcircle of ",[27,2880,2882,2903],{"className":2881},[30],[27,2883,2885],{"className":2884},[34],[36,2886,2887],{"xmlns":38},[40,2888,2889,2900],{},[43,2890,2891,2894,2896,2898],{},[46,2892,2893],{"mathvariant":2299},"△",[46,2895,149],{},[46,2897,179],{},[46,2899,210],{},[50,2901,2902],{"encoding":52},"\\triangle ABC",[27,2904,2906],{"className":2905,"ariaHidden":57},[56],[27,2907,2909,2912,2915,2918,2921],{"className":2908},[61],[27,2910],{"className":2911,"style":523},[65],[27,2913,2893],{"className":2914},[70],[27,2916,149],{"className":2917},[70,71],[27,2919,179],{"className":2920,"style":194},[70,71],[27,2922,210],{"className":2923,"style":225},[70,71],"). If ",[27,2926,2928,2941],{"className":2927},[30],[27,2929,2931],{"className":2930},[34],[36,2932,2933],{"xmlns":38},[40,2934,2935,2939],{},[43,2936,2937],{},[46,2938,48],{},[50,2940,48],{"encoding":52},[27,2942,2944],{"className":2943,"ariaHidden":57},[56],[27,2945,2947,2950],{"className":2946},[61],[27,2948],{"className":2949,"style":66},[65],[27,2951,48],{"className":2952,"style":72},[70,71]," lies on that circle, infinitely many positions produce the same observed angles (inscribed angle theorem).",[15,2955,2956],{},"Algebraically, barycentric denominators approach zero:",[15,2958,2959],{},[27,2960,2962,3037],{"className":2961},[30],[27,2963,2965],{"className":2964},[34],[36,2966,2967],{"xmlns":38},[40,2968,2969,3034],{},[43,2970,2971,2974,2977,2979,2982,2984,2986,2988,2991,2993,2996,2998,3000,3002,3004,3006,3008,3010,3012,3014,3016,3018,3020,3022,3024,3026,3028,3030,3032],{},[46,2972,2973],{},"cot",[746,2975,2976],{},"⁡",[46,2978,149],{},[746,2980,2981],{},"−",[46,2983,2973],{},[746,2985,2976],{},[46,2987,2477],{},[746,2989,2990],{},"→",[508,2992,2843],{},[878,2994],{"width":2995},"2em",[46,2997,2973],{},[746,2999,2976],{},[46,3001,179],{},[746,3003,2981],{},[46,3005,2973],{},[746,3007,2976],{},[46,3009,2610],{},[746,3011,2990],{},[508,3013,2843],{},[878,3015],{"width":2995},[46,3017,2973],{},[746,3019,2976],{},[46,3021,210],{},[746,3023,2981],{},[46,3025,2973],{},[746,3027,2976],{},[46,3029,2338],{},[746,3031,2990],{},[508,3033,2843],{},[50,3035,3036],{"encoding":52},"\\cot A - \\cot\\beta \\to 0 \\qquad \\cot B - \\cot\\gamma \\to 0 \\qquad \\cot C - \\cot\\alpha \\to 0",[27,3038,3040,3067,3091,3125,3150,3183,3208],{"className":3039,"ariaHidden":57},[56],[27,3041,3043,3047,3051,3055,3058,3061,3064],{"className":3042},[61],[27,3044],{"className":3045,"style":3046},[65],"height:0.7667em;vertical-align:-0.0833em;",[27,3048,2973],{"className":3049},[3050],"mop",[27,3052],{"className":3053,"style":3054},[878],"margin-right:0.1667em;",[27,3056,149],{"className":3057},[70,71],[27,3059],{"className":3060,"style":2783},[878],[27,3062,2981],{"className":3063},[977],[27,3065],{"className":3066,"style":2783},[878],[27,3068,3070,3073,3076,3079,3082,3085,3088],{"className":3069},[61],[27,3071],{"className":3072,"style":523},[65],[27,3074,2973],{"className":3075},[3050],[27,3077],{"className":3078,"style":3054},[878],[27,3080,2477],{"className":3081,"style":2493},[70,71],[27,3083],{"className":3084,"style":879},[878],[27,3086,2990],{"className":3087},[883],[27,3089],{"className":3090,"style":879},[878],[27,3092,3094,3097,3100,3104,3107,3110,3113,3116,3119,3122],{"className":3093},[61],[27,3095],{"className":3096,"style":3046},[65],[27,3098,2843],{"className":3099},[70],[27,3101],{"className":3102,"style":3103},[878],"margin-right:2em;",[27,3105],{"className":3106,"style":3054},[878],[27,3108,2973],{"className":3109},[3050],[27,3111],{"className":3112,"style":3054},[878],[27,3114,179],{"className":3115,"style":194},[70,71],[27,3117],{"className":3118,"style":2783},[878],[27,3120,2981],{"className":3121},[977],[27,3123],{"className":3124,"style":2783},[878],[27,3126,3128,3132,3135,3138,3141,3144,3147],{"className":3127},[61],[27,3129],{"className":3130,"style":3131},[65],"height:0.8095em;vertical-align:-0.1944em;",[27,3133,2973],{"className":3134},[3050],[27,3136],{"className":3137,"style":3054},[878],[27,3139,2610],{"className":3140,"style":2626},[70,71],[27,3142],{"className":3143,"style":879},[878],[27,3145,2990],{"className":3146},[883],[27,3148],{"className":3149,"style":879},[878],[27,3151,3153,3156,3159,3162,3165,3168,3171,3174,3177,3180],{"className":3152},[61],[27,3154],{"className":3155,"style":3046},[65],[27,3157,2843],{"className":3158},[70],[27,3160],{"className":3161,"style":3103},[878],[27,3163],{"className":3164,"style":3054},[878],[27,3166,2973],{"className":3167},[3050],[27,3169],{"className":3170,"style":3054},[878],[27,3172,210],{"className":3173,"style":225},[70,71],[27,3175],{"className":3176,"style":2783},[878],[27,3178,2981],{"className":3179},[977],[27,3181],{"className":3182,"style":2783},[878],[27,3184,3186,3190,3193,3196,3199,3202,3205],{"className":3185},[61],[27,3187],{"className":3188,"style":3189},[65],"height:0.6151em;",[27,3191,2973],{"className":3192},[3050],[27,3194],{"className":3195,"style":3054},[878],[27,3197,2338],{"className":3198,"style":2354},[70,71],[27,3200],{"className":3201,"style":879},[878],[27,3203,2990],{"className":3204},[883],[27,3206],{"className":3207,"style":879},[878],[27,3209,3211,3215],{"className":3210},[61],[27,3212],{"className":3213,"style":3214},[65],"height:0.6444em;",[27,3216,2843],{"className":3217},[70],[88,3219,3221],{"id":3220},"ensuring-geometric-strength","Ensuring geometric strength",[2319,3223,3224,3302,3474],{},[2322,3225,3226,2143,3229,3257,3258,3301],{},[22,3227,3228],{},"Enclosure:",[27,3230,3232,3245],{"className":3231},[30],[27,3233,3235],{"className":3234},[34],[36,3236,3237],{"xmlns":38},[40,3238,3239,3243],{},[43,3240,3241],{},[46,3242,48],{},[50,3244,48],{"encoding":52},[27,3246,3248],{"className":3247,"ariaHidden":57},[56],[27,3249,3251,3254],{"className":3250},[61],[27,3252],{"className":3253,"style":66},[65],[27,3255,48],{"className":3256,"style":72},[70,71]," inside ",[27,3259,3261,3280],{"className":3260},[30],[27,3262,3264],{"className":3263},[34],[36,3265,3266],{"xmlns":38},[40,3267,3268,3278],{},[43,3269,3270,3272,3274,3276],{},[46,3271,2893],{"mathvariant":2299},[46,3273,149],{},[46,3275,179],{},[46,3277,210],{},[50,3279,2902],{"encoding":52},[27,3281,3283],{"className":3282,"ariaHidden":57},[56],[27,3284,3286,3289,3292,3295,3298],{"className":3285},[61],[27,3287],{"className":3288,"style":523},[65],[27,3290,2893],{"className":3291},[70],[27,3293,149],{"className":3294},[70,71],[27,3296,179],{"className":3297,"style":194},[70,71],[27,3299,210],{"className":3300,"style":225},[70,71]," is most stable (positive weights, robust to noise).",[2322,3303,3304,3307,3308,3336,3337,3365,3366,3394,3395,3423,3424,3473],{},[22,3305,3306],{},"Outside triangle:"," avoid placing ",[27,3309,3311,3324],{"className":3310},[30],[27,3312,3314],{"className":3313},[34],[36,3315,3316],{"xmlns":38},[40,3317,3318,3322],{},[43,3319,3320],{},[46,3321,48],{},[50,3323,48],{"encoding":52},[27,3325,3327],{"className":3326,"ariaHidden":57},[56],[27,3328,3330,3333],{"className":3329},[61],[27,3331],{"className":3332,"style":66},[65],[27,3334,48],{"className":3335,"style":72},[70,71]," on or near the circumcircle. A quick check: if interior angle at ",[27,3338,3340,3353],{"className":3339},[30],[27,3341,3343],{"className":3342},[34],[36,3344,3345],{"xmlns":38},[40,3346,3347,3351],{},[43,3348,3349],{},[46,3350,179],{},[50,3352,179],{"encoding":52},[27,3354,3356],{"className":3355,"ariaHidden":57},[56],[27,3357,3359,3362],{"className":3358},[61],[27,3360],{"className":3361,"style":66},[65],[27,3363,179],{"className":3364,"style":194},[70,71]," plus observed ",[27,3367,3369,3382],{"className":3368},[30],[27,3370,3372],{"className":3371},[34],[36,3373,3374],{"xmlns":38},[40,3375,3376,3380],{},[43,3377,3378],{},[46,3379,2610],{},[50,3381,2613],{"encoding":52},[27,3383,3385],{"className":3384,"ariaHidden":57},[56],[27,3386,3388,3391],{"className":3387},[61],[27,3389],{"className":3390,"style":2244},[65],[27,3392,2610],{"className":3393,"style":2626},[70,71]," (opposite ",[27,3396,3398,3411],{"className":3397},[30],[27,3399,3401],{"className":3400},[34],[36,3402,3403],{"xmlns":38},[40,3404,3405,3409],{},[43,3406,3407],{},[46,3408,179],{},[50,3410,179],{"encoding":52},[27,3412,3414],{"className":3413,"ariaHidden":57},[56],[27,3415,3417,3420],{"className":3416},[61],[27,3418],{"className":3419,"style":66},[65],[27,3421,179],{"className":3422,"style":194},[70,71],") ",[27,3425,3427,3447],{"className":3426},[30],[27,3428,3430],{"className":3429},[34],[36,3431,3432],{"xmlns":38},[40,3433,3434,3444],{},[43,3435,3436,3439,3442],{},[746,3437,3438],{},"≈",[508,3440,3441],{},"180",[46,3443,2300],{"mathvariant":2299},[50,3445,3446],{"encoding":52},"\\approx 180°",[27,3448,3450,3463],{"className":3449,"ariaHidden":57},[56],[27,3451,3453,3457,3460],{"className":3452},[61],[27,3454],{"className":3455,"style":3456},[65],"height:0.4831em;",[27,3458,3438],{"className":3459},[883],[27,3461],{"className":3462,"style":879},[878],[27,3464,3466,3469],{"className":3465},[61],[27,3467],{"className":3468,"style":2313},[65],[27,3470,3472],{"className":3471},[70],"180°",", you are near the danger circle.",[2322,3475,3476,3479],{},[22,3477,3478],{},"Collinearity:"," three collinear control points give degenerate geometry — extremely unstable.",[3481,3482,3487],"pre",{"className":3483,"code":3485,"language":3486},[3484],"language-text","              A -------- B\n             \u002F    *P?    \\\n            \u002F   (danger    \\\n           C --- circle) --- \n","text",[3488,3489,3485],"code",{"__ignoreMap":3490},"",[10,3492,3494],{"id":3493},"field-observation-practice","Field Observation Practice",[88,3496,3498],{"id":3497},"face-i-and-face-ii","Face I and Face II",[15,3500,3501,3502,3505,3506,3509],{},"Eliminate collimation, tilt-axis, and eccentricity errors by observing each target in ",[22,3503,3504],{},"Face I"," (direct) and ",[22,3507,3508],{},"Face II"," (reverse) and averaging.",[88,3511,3513],{"id":3512},"horizon-closure","Horizon closure",[15,3515,3516,3517,3612,3613,2317],{},"After sighting ",[27,3518,3520,3546],{"className":3519},[30],[27,3521,3523],{"className":3522},[34],[36,3524,3525],{"xmlns":38},[40,3526,3527,3543],{},[43,3528,3529,3531,3533,3535,3537,3539,3541],{},[46,3530,149],{},[746,3532,2990],{},[46,3534,179],{},[746,3536,2990],{},[46,3538,210],{},[746,3540,2990],{},[46,3542,149],{},[50,3544,3545],{"encoding":52},"A \\to B \\to C \\to A",[27,3547,3549,3567,3585,3603],{"className":3548,"ariaHidden":57},[56],[27,3550,3552,3555,3558,3561,3564],{"className":3551},[61],[27,3553],{"className":3554,"style":66},[65],[27,3556,149],{"className":3557},[70,71],[27,3559],{"className":3560,"style":879},[878],[27,3562,2990],{"className":3563},[883],[27,3565],{"className":3566,"style":879},[878],[27,3568,3570,3573,3576,3579,3582],{"className":3569},[61],[27,3571],{"className":3572,"style":66},[65],[27,3574,179],{"className":3575,"style":194},[70,71],[27,3577],{"className":3578,"style":879},[878],[27,3580,2990],{"className":3581},[883],[27,3583],{"className":3584,"style":879},[878],[27,3586,3588,3591,3594,3597,3600],{"className":3587},[61],[27,3589],{"className":3590,"style":66},[65],[27,3592,210],{"className":3593,"style":225},[70,71],[27,3595],{"className":3596,"style":879},[878],[27,3598,2990],{"className":3599},[883],[27,3601],{"className":3602,"style":879},[878],[27,3604,3606,3609],{"className":3605},[61],[27,3607],{"className":3608,"style":66},[65],[27,3610,149],{"className":3611},[70,71],", compare the angle sum to 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3",[27,4307,4309,4327],{"className":4308,"ariaHidden":57},[56],[27,4310,4312,4315,4318,4321,4324],{"className":4311},[61],[27,4313],{"className":4314,"style":2213},[65],[27,4316,4171],{"className":4317},[70,71],[27,4319],{"className":4320,"style":879},[878],[27,4322,748],{"className":4323},[883],[27,4325],{"className":4326,"style":879},[878],[27,4328,4330,4333],{"className":4329},[61],[27,4331],{"className":4332,"style":3214},[65],[27,4334,670],{"className":4335},[70]," angles.",[88,4338,4340],{"id":4339},"height","Height",[15,4342,4343],{},"Standard Tienstra resection is planimetric (2D). Record instrument and target heights for pairing with later vertical observations.",[10,4345,4347],{"id":4346},"computation-walkthrough","Computation Walkthrough",[88,4349,4351],{"id":4350},"step-1-interior-angles-of-the-control-triangle","Step 1: Interior angles of the control triangle",[15,4353,4354,4355,4359,4360,164,4388,164,4416,4444,4445,2317],{},"Compute clockwise grid bearings between control points (use the ",[75,4356,4358],{"href":4357},"\u002Ftools\u002Fbearing-distance","Bearing & distance calculator"," for inverse checks). For CCW-ordered ",[27,4361,4363,4376],{"className":4362},[30],[27,4364,4366],{"className":4365},[34],[36,4367,4368],{"xmlns":38},[40,4369,4370,4374],{},[43,4371,4372],{},[46,4373,149],{},[50,4375,149],{"encoding":52},[27,4377,4379],{"className":4378,"ariaHidden":57},[56],[27,4380,4382,4385],{"className":4381},[61],[27,4383],{"className":4384,"style":66},[65],[27,4386,149],{"className":4387},[70,71],[27,4389,4391,4404],{"className":4390},[30],[27,4392,4394],{"className":4393},[34],[36,4395,4396],{"xmlns":38},[40,4397,4398,4402],{},[43,4399,4400],{},[46,4401,179],{},[50,4403,179],{"encoding":52},[27,4405,4407],{"className":4406,"ariaHidden":57},[56],[27,4408,4410,4413],{"className":4409},[61],[27,4411],{"className":4412,"style":66},[65],[27,4414,179],{"className":4415,"style":194},[70,71],[27,4417,4419,4432],{"className":4418},[30],[27,4420,4422],{"className":4421},[34],[36,4423,4424],{"xmlns":38},[40,4425,4426,4430],{},[43,4427,4428],{},[46,4429,210],{},[50,4431,210],{"encoding":52},[27,4433,4435],{"className":4434,"ariaHidden":57},[56],[27,4436,4438,4441],{"className":4437},[61],[27,4439],{"className":4440,"style":66},[65],[27,4442,210],{"className":4443,"style":225},[70,71],", interior angle at ",[27,4446,4448,4461],{"className":4447},[30],[27,4449,4451],{"className":4450},[34],[36,4452,4453],{"xmlns":38},[40,4454,4455,4459],{},[43,4456,4457],{},[46,4458,149],{},[50,4460,149],{"encoding":52},[27,4462,4464],{"className":4463,"ariaHidden":57},[56],[27,4465,4467,4470],{"className":4466},[61],[27,4468],{"className":4469,"style":66},[65],[27,4471,149],{"className":4472},[70,71],[15,4474,4475],{},[27,4476,4478,4516],{"className":4477},[30],[27,4479,4481],{"className":4480},[34],[36,4482,4483],{"xmlns":38},[40,4484,4485,4513],{},[43,4486,4487,4489,4491,4501,4503],{},[46,4488,149],{},[746,4490,748],{},[502,4492,4493,4495],{},[46,4494,3664],{},[43,4496,4497,4499],{},[46,4498,149],{},[46,4500,179],{},[746,4502,2981],{},[502,4504,4505,4507],{},[46,4506,3664],{},[43,4508,4509,4511],{},[46,4510,149],{},[46,4512,210],{},[50,4514,4515],{"encoding":52},"A = \\theta_{AB} - \\theta_{AC}",[27,4517,4519,4537,4599],{"className":4518,"ariaHidden":57},[56],[27,4520,4522,4525,4528,4531,4534],{"className":4521},[61],[27,4523],{"className":4524,"style":66},[65],[27,4526,149],{"className":4527},[70,71],[27,4529],{"className":4530,"style":879},[878],[27,4532,748],{"className":4533},[883],[27,4535],{"className":4536,"style":879},[878],[27,4538,4540,4543,4590,4593,4596],{"className":4539},[61],[27,4541],{"className":4542,"style":3703},[65],[27,4544,4546,4549],{"className":4545},[70],[27,4547,3664],{"className":4548,"style":3736},[70,71],[27,4550,4552],{"className":4551},[534],[27,4553,4555,4582],{"className":4554},[538,539],[27,4556,4558,4579],{"className":4557},[543],[27,4559,4561],{"className":4560,"style":851},[547],[27,4562,4564,4567],{"style":4563},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[27,4565],{"className":4566,"style":556},[555],[27,4568,4570],{"className":4569},[560,561,562,563],[27,4571,4573,4576],{"className":4572},[70,563],[27,4574,149],{"className":4575},[70,71,563],[27,4577,179],{"className":4578,"style":194},[70,71,563],[27,4580,571],{"className":4581},[570],[27,4583,4585],{"className":4584},[543],[27,4586,4588],{"className":4587,"style":578},[547],[27,4589],{},[27,4591],{"className":4592,"style":2783},[878],[27,4594,2981],{"className":4595},[977],[27,4597],{"className":4598,"style":2783},[878],[27,4600,4602,4605],{"className":4601},[61],[27,4603],{"className":4604,"style":3703},[65],[27,4606,4608,4611],{"className":4607},[70],[27,4609,3664],{"className":4610,"style":3736},[70,71],[27,4612,4614],{"className":4613},[534],[27,4615,4617,4643],{"className":4616},[538,539],[27,4618,4620,4640],{"className":4619},[543],[27,4621,4623],{"className":4622,"style":851},[547],[27,4624,4625,4628],{"style":4563},[27,4626],{"className":4627,"style":556},[555],[27,4629,4631],{"className":4630},[560,561,562,563],[27,4632,4634,4637],{"className":4633},[70,563],[27,4635,149],{"className":4636},[70,71,563],[27,4638,210],{"className":4639,"style":225},[70,71,563],[27,4641,571],{"className":4642},[570],[27,4644,4646],{"className":4645},[543],[27,4647,4649],{"className":4648,"style":578},[547],[27,4650],{},[15,4652,4653,4654,4685,4686,4716,4717,4745,4746,4774,4775,485],{},"(normalize to ",[27,4655,4657,4673],{"className":4656},[30],[27,4658,4660],{"className":4659},[34],[36,4661,4662],{"xmlns":38},[40,4663,4664,4670],{},[43,4665,4666,4668],{},[508,4667,2843],{},[46,4669,2300],{"mathvariant":2299},[50,4671,4672],{"encoding":52},"0°",[27,4674,4676],{"className":4675,"ariaHidden":57},[56],[27,4677,4679,4682],{"className":4678},[61],[27,4680],{"className":4681,"style":2313},[65],[27,4683,4672],{"className":4684},[70],"–",[27,4687,4689,4704],{"className":4688},[30],[27,4690,4692],{"className":4691},[34],[36,4693,4694],{"xmlns":38},[40,4695,4696,4702],{},[43,4697,4698,4700],{},[508,4699,2296],{},[46,4701,2300],{"mathvariant":2299},[50,4703,2303],{"encoding":52},[27,4705,4707],{"className":4706,"ariaHidden":57},[56],[27,4708,4710,4713],{"className":4709},[61],[27,4711],{"className":4712,"style":2313},[65],[27,4714,2303],{"className":4715},[70],"). Similarly find ",[27,4718,4720,4733],{"className":4719},[30],[27,4721,4723],{"className":4722},[34],[36,4724,4725],{"xmlns":38},[40,4726,4727,4731],{},[43,4728,4729],{},[46,4730,179],{},[50,4732,179],{"encoding":52},[27,4734,4736],{"className":4735,"ariaHidden":57},[56],[27,4737,4739,4742],{"className":4738},[61],[27,4740],{"className":4741,"style":66},[65],[27,4743,179],{"className":4744,"style":194},[70,71]," and ",[27,4747,4749,4762],{"className":4748},[30],[27,4750,4752],{"className":4751},[34],[36,4753,4754],{"xmlns":38},[40,4755,4756,4760],{},[43,4757,4758],{},[46,4759,210],{},[50,4761,210],{"encoding":52},[27,4763,4765],{"className":4764,"ariaHidden":57},[56],[27,4766,4768,4771],{"className":4767},[61],[27,4769],{"className":4770,"style":66},[65],[27,4772,210],{"className":4773,"style":225},[70,71],". Verify ",[27,4776,4778,4806],{"className":4777},[30],[27,4779,4781],{"className":4780},[34],[36,4782,4783],{"xmlns":38},[40,4784,4785,4803],{},[43,4786,4787,4789,4791,4793,4795,4797,4799,4801],{},[46,4788,149],{},[746,4790,768],{},[46,4792,179],{},[746,4794,768],{},[46,4796,210],{},[746,4798,748],{},[508,4800,3441],{},[46,4802,2300],{"mathvariant":2299},[50,4804,4805],{"encoding":52},"A + B + C = 180°",[27,4807,4809,4827,4845,4863],{"className":4808,"ariaHidden":57},[56],[27,4810,4812,4815,4818,4821,4824],{"className":4811},[61],[27,4813],{"className":4814,"style":3046},[65],[27,4816,149],{"className":4817},[70,71],[27,4819],{"className":4820,"style":2783},[878],[27,4822,768],{"className":4823},[977],[27,4825],{"className":4826,"style":2783},[878],[27,4828,4830,4833,4836,4839,4842],{"className":4829},[61],[27,4831],{"className":4832,"style":3046},[65],[27,4834,179],{"className":4835,"style":194},[70,71],[27,4837],{"className":4838,"style":2783},[878],[27,4840,768],{"className":4841},[977],[27,4843],{"className":4844,"style":2783},[878],[27,4846,4848,4851,4854,4857,4860],{"className":4847},[61],[27,4849],{"className":4850,"style":66},[65],[27,4852,210],{"className":4853,"style":225},[70,71],[27,4855],{"className":4856,"style":879},[878],[27,4858,748],{"className":4859},[883],[27,4861],{"className":4862,"style":879},[878],[27,4864,4866,4869],{"className":4865},[61],[27,4867],{"className":4868,"style":2313},[65],[27,4870,3472],{"className":4871},[70],[88,4873,4875],{"id":4874},"step-2-barycentric-weights","Step 2: Barycentric weights",[15,4877,4878,4879,4907,4908,4936,4937,164,4965,4936,4993,164,5021,4936,5049,5077],{},"With observed angles at ",[27,4880,4882,4895],{"className":4881},[30],[27,4883,4885],{"className":4884},[34],[36,4886,4887],{"xmlns":38},[40,4888,4889,4893],{},[43,4890,4891],{},[46,4892,48],{},[50,4894,48],{"encoding":52},[27,4896,4898],{"className":4897,"ariaHidden":57},[56],[27,4899,4901,4904],{"className":4900},[61],[27,4902],{"className":4903,"style":66},[65],[27,4905,48],{"className":4906,"style":72},[70,71]," (",[27,4909,4911,4924],{"className":4910},[30],[27,4912,4914],{"className":4913},[34],[36,4915,4916],{"xmlns":38},[40,4917,4918,4922],{},[43,4919,4920],{},[46,4921,2338],{},[50,4923,2341],{"encoding":52},[27,4925,4927],{"className":4926,"ariaHidden":57},[56],[27,4928,4930,4933],{"className":4929},[61],[27,4931],{"className":4932,"style":2213},[65],[27,4934,2338],{"className":4935,"style":2354},[70,71]," opposite ",[27,4938,4940,4953],{"className":4939},[30],[27,4941,4943],{"className":4942},[34],[36,4944,4945],{"xmlns":38},[40,4946,4947,4951],{},[43,4948,4949],{},[46,4950,210],{},[50,4952,210],{"encoding":52},[27,4954,4956],{"className":4955,"ariaHidden":57},[56],[27,4957,4959,4962],{"className":4958},[61],[27,4960],{"className":4961,"style":66},[65],[27,4963,210],{"className":4964,"style":225},[70,71],[27,4966,4968,4981],{"className":4967},[30],[27,4969,4971],{"className":4970},[34],[36,4972,4973],{"xmlns":38},[40,4974,4975,4979],{},[43,4976,4977],{},[46,4978,2477],{},[50,4980,2480],{"encoding":52},[27,4982,4984],{"className":4983,"ariaHidden":57},[56],[27,4985,4987,4990],{"className":4986},[61],[27,4988],{"className":4989,"style":523},[65],[27,4991,2477],{"className":4992,"style":2493},[70,71],[27,4994,4996,5009],{"className":4995},[30],[27,4997,4999],{"className":4998},[34],[36,5000,5001],{"xmlns":38},[40,5002,5003,5007],{},[43,5004,5005],{},[46,5006,149],{},[50,5008,149],{"encoding":52},[27,5010,5012],{"className":5011,"ariaHidden":57},[56],[27,5013,5015,5018],{"className":5014},[61],[27,5016],{"className":5017,"style":66},[65],[27,5019,149],{"className":5020},[70,71],[27,5022,5024,5037],{"className":5023},[30],[27,5025,5027],{"className":5026},[34],[36,5028,5029],{"xmlns":38},[40,5030,5031,5035],{},[43,5032,5033],{},[46,5034,2610],{},[50,5036,2613],{"encoding":52},[27,5038,5040],{"className":5039,"ariaHidden":57},[56],[27,5041,5043,5046],{"className":5042},[61],[27,5044],{"className":5045,"style":2244},[65],[27,5047,2610],{"className":5048,"style":2626},[70,71],[27,5050,5052,5065],{"className":5051},[30],[27,5053,5055],{"className":5054},[34],[36,5056,5057],{"xmlns":38},[40,5058,5059,5063],{},[43,5060,5061],{},[46,5062,179],{},[50,5064,179],{"encoding":52},[27,5066,5068],{"className":5067,"ariaHidden":57},[56],[27,5069,5071,5074],{"className":5070},[61],[27,5072],{"className":5073,"style":66},[65],[27,5075,179],{"className":5076,"style":194},[70,71],"):",[15,5079,5080],{},[27,5081,5083,5123],{"className":5082},[30],[27,5084,5086],{"className":5085},[34],[36,5087,5088],{"xmlns":38},[40,5089,5090,5120],{},[43,5091,5092,5098,5100],{},[502,5093,5094,5096],{},[46,5095,506],{},[508,5097,510],{},[746,5099,748],{},[750,5101,5102,5104],{},[508,5103,510],{},[43,5105,5106,5108,5110,5112,5114,5116,5118],{},[46,5107,2973],{},[746,5109,2976],{},[46,5111,149],{},[746,5113,2981],{},[46,5115,2973],{},[746,5117,2976],{},[46,5119,2477],{},[50,5121,5122],{"encoding":52},"f_1 = \\frac{1}{\\cot A - \\cot \\beta}",[27,5124,5126,5181],{"className":5125,"ariaHidden":57},[56],[27,5127,5129,5132,5172,5175,5178],{"className":5128},[61],[27,5130],{"className":5131,"style":523},[65],[27,5133,5135,5138],{"className":5134},[70],[27,5136,506],{"className":5137,"style":530},[70,71],[27,5139,5141],{"className":5140},[534],[27,5142,5144,5164],{"className":5143},[538,539],[27,5145,5147,5161],{"className":5146},[543],[27,5148,5150],{"className":5149,"style":548},[547],[27,5151,5152,5155],{"style":551},[27,5153],{"className":5154,"style":556},[555],[27,5156,5158],{"className":5157},[560,561,562,563],[27,5159,510],{"className":5160},[70,563],[27,5162,571],{"className":5163},[570],[27,5165,5167],{"className":5166},[543],[27,5168,5170],{"className":5169,"style":578},[547],[27,5171],{},[27,5173],{"className":5174,"style":879},[878],[27,5176,748],{"className":5177},[883],[27,5179],{"className":5180,"style":879},[878],[27,5182,5184,5188],{"className":5183},[61],[27,5185],{"className":5186,"style":5187},[65],"height:1.3262em;vertical-align:-0.4811em;",[27,5189,5191,5194,5294],{"className":5190},[70],[27,5192],{"className":5193},[900,901],[27,5195,5197],{"className":5196},[750],[27,5198,5200,5286],{"className":5199},[538,539],[27,5201,5203,5283],{"className":5202},[543],[27,5204,5207,5260,5268],{"className":5205,"style":5206},[547],"height:0.8451em;",[27,5208,5209,5212],{"style":917},[27,5210],{"className":5211,"style":921},[555],[27,5213,5215],{"className":5214},[560,561,562,563],[27,5216,5218,5232,5236,5239,5242,5254,5257],{"className":5217},[70,563],[27,5219,5221,5224,5228],{"className":5220},[3050,563],[27,5222,4165],{"className":5223},[563],[27,5225,5227],{"className":5226},[563],"o",[27,5229,5231],{"className":5230},[563],"t",[27,5233],{"className":5234,"style":5235},[878,563],"margin-right:0.1952em;",[27,5237,149],{"className":5238},[70,71,563],[27,5240,2981],{"className":5241},[977,563],[27,5243,5245,5248,5251],{"className":5244},[3050,563],[27,5246,4165],{"className":5247},[563],[27,5249,5227],{"className":5250},[563],[27,5252,5231],{"className":5253},[563],[27,5255],{"className":5256,"style":5235},[878,563],[27,5258,2477],{"className":5259,"style":2493},[70,71,563],[27,5261,5262,5265],{"style":1063},[27,5263],{"className":5264,"style":921},[555],[27,5266],{"className":5267,"style":1071},[1070],[27,5269,5271,5274],{"style":5270},"top:-3.394em;",[27,5272],{"className":5273,"style":921},[555],[27,5275,5277],{"className":5276},[560,561,562,563],[27,5278,5280],{"className":5279},[70,563],[27,5281,510],{"className":5282},[70,563],[27,5284,571],{"className":5285},[570],[27,5287,5289],{"className":5288},[543],[27,5290,5292],{"className":5291,"style":1342},[547],[27,5293],{},[27,5295],{"className":5296},[1348,901],[15,5298,5299],{},[27,5300,5302,5342],{"className":5301},[30],[27,5303,5305],{"className":5304},[34],[36,5306,5307],{"xmlns":38},[40,5308,5309,5339],{},[43,5310,5311,5317,5319],{},[502,5312,5313,5315],{},[46,5314,506],{},[508,5316,599],{},[746,5318,748],{},[750,5320,5321,5323],{},[508,5322,510],{},[43,5324,5325,5327,5329,5331,5333,5335,5337],{},[46,5326,2973],{},[746,5328,2976],{},[46,5330,179],{},[746,5332,2981],{},[46,5334,2973],{},[746,5336,2976],{},[46,5338,2610],{},[50,5340,5341],{"encoding":52},"f_2 = \\frac{1}{\\cot B - \\cot \\gamma}",[27,5343,5345,5400],{"className":5344,"ariaHidden":57},[56],[27,5346,5348,5351,5391,5394,5397],{"className":5347},[61],[27,5349],{"className":5350,"style":523},[65],[27,5352,5354,5357],{"className":5353},[70],[27,5355,506],{"className":5356,"style":530},[70,71],[27,5358,5360],{"className":5359},[534],[27,5361,5363,5383],{"className":5362},[538,539],[27,5364,5366,5380],{"className":5365},[543],[27,5367,5369],{"className":5368,"style":548},[547],[27,5370,5371,5374],{"style":551},[27,5372],{"className":5373,"style":556},[555],[27,5375,5377],{"className":5376},[560,561,562,563],[27,5378,599],{"className":5379},[70,563],[27,5381,571],{"className":5382},[570],[27,5384,5386],{"className":5385},[543],[27,5387,5389],{"className":5388,"style":578},[547],[27,5390],{},[27,5392],{"className":5393,"style":879},[878],[27,5395,748],{"className":5396},[883],[27,5398],{"className":5399,"style":879},[878],[27,5401,5403,5406],{"className":5402},[61],[27,5404],{"className":5405,"style":5187},[65],[27,5407,5409,5412,5507],{"className":5408},[70],[27,5410],{"className":5411},[900,901],[27,5413,5415],{"className":5414},[750],[27,5416,5418,5499],{"className":5417},[538,539],[27,5419,5421,5496],{"className":5420},[543],[27,5422,5424,5474,5482],{"className":5423,"style":5206},[547],[27,5425,5426,5429],{"style":917},[27,5427],{"className":5428,"style":921},[555],[27,5430,5432],{"className":5431},[560,561,562,563],[27,5433,5435,5447,5450,5453,5456,5468,5471],{"className":5434},[70,563],[27,5436,5438,5441,5444],{"className":5437},[3050,563],[27,5439,4165],{"className":5440},[563],[27,5442,5227],{"className":5443},[563],[27,5445,5231],{"className":5446},[563],[27,5448],{"className":5449,"style":5235},[878,563],[27,5451,179],{"className":5452,"style":194},[70,71,563],[27,5454,2981],{"className":5455},[977,563],[27,5457,5459,5462,5465],{"className":5458},[3050,563],[27,5460,4165],{"className":5461},[563],[27,5463,5227],{"className":5464},[563],[27,5466,5231],{"className":5467},[563],[27,5469],{"className":5470,"style":5235},[878,563],[27,5472,2610],{"className":5473,"style":2626},[70,71,563],[27,5475,5476,5479],{"style":1063},[27,5477],{"className":5478,"style":921},[555],[27,5480],{"className":5481,"style":1071},[1070],[27,5483,5484,5487],{"style":5270},[27,5485],{"className":5486,"style":921},[555],[27,5488,5490],{"className":5489},[560,561,562,563],[27,5491,5493],{"className":5492},[70,563],[27,5494,510],{"className":5495},[70,563],[27,5497,571],{"className":5498},[570],[27,5500,5502],{"className":5501},[543],[27,5503,5505],{"className":5504,"style":1342},[547],[27,5506],{},[27,5508],{"className":5509},[1348,901],[15,5511,5512],{},[27,5513,5515,5555],{"className":5514},[30],[27,5516,5518],{"className":5517},[34],[36,5519,5520],{"xmlns":38},[40,5521,5522,5552],{},[43,5523,5524,5530,5532],{},[502,5525,5526,5528],{},[46,5527,506],{},[508,5529,670],{},[746,5531,748],{},[750,5533,5534,5536],{},[508,5535,510],{},[43,5537,5538,5540,5542,5544,5546,5548,5550],{},[46,5539,2973],{},[746,5541,2976],{},[46,5543,210],{},[746,5545,2981],{},[46,5547,2973],{},[746,5549,2976],{},[46,5551,2338],{},[50,5553,5554],{"encoding":52},"f_3 = \\frac{1}{\\cot C - \\cot 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3: Coordinates of 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= \\frac{f_1 E_A + f_2 E_B + f_3 E_C}{f_1 + f_2 + f_3} \\qquad N_P = \\frac{f_1 N_A + f_2 N_B + f_3 N_C}{f_1 + f_2 + f_3}",[27,5929,5931,5986,6478],{"className":5930,"ariaHidden":57},[56],[27,5932,5934,5937,5977,5980,5983],{"className":5933},[61],[27,5935],{"className":5936,"style":831},[65],[27,5938,5940,5943],{"className":5939},[70],[27,5941,742],{"className":5942,"style":838},[70,71],[27,5944,5946],{"className":5945},[534],[27,5947,5949,5969],{"className":5948},[538,539],[27,5950,5952,5966],{"className":5951},[543],[27,5953,5955],{"className":5954,"style":851},[547],[27,5956,5957,5960],{"style":854},[27,5958],{"className":5959,"style":556},[555],[27,5961,5963],{"className":5962},[560,561,562,563],[27,5964,48],{"className":5965,"style":72},[70,71,563],[27,5967,571],{"className":5968},[570],[27,5970,5972],{"className":5971},[543],[27,5973,5975],{"className":5974,"style":578},[547],[27,5976],{},[27,5978],{"className":5979,"style":879},[878],[27,5981,748],{"className":5982},[883],[27,5984],{"className":5985,"style":879},[878],[27,5987,5989,5992,6426,6429,6469,6472,6475],{"className":5988},[61],[27,5990],{"className":5991,"style":893},[65],[27,5993,5995,5998,6423],{"className":5994},[70],[27,5996],{"className":5997},[900,901],[27,5999,6001],{"className":6000},[750],[27,6002,6004,6415],{"className":6003},[538,539],[27,6005,6007,6412],{"className":6006},[543],[27,6008,6010,6147,6155],{"className":6009,"style":914},[547],[27,6011,6012,6015],{"style":917},[27,6013],{"className":6014,"style":921},[555],[27,6016,6018],{"className":6017},[560,561,562,563],[27,6019,6021,6061,6064,6104,6107],{"className":6020},[70,563],[27,6022,6024,6027],{"className":6023},[70,563],[27,6025,506],{"className":6026,"style":530},[70,71,563],[27,6028,6030],{"className":6029},[534],[27,6031,6033,6053],{"className":6032},[538,539],[27,6034,6036,6050],{"className":6035},[543],[27,6037,6039],{"className":6038,"style":946},[547],[27,6040,6041,6044],{"style":949},[27,6042],{"className":6043,"style":953},[555],[27,6045,6047],{"className":6046},[560,957,958,563],[27,6048,510],{"className":6049},[70,563],[27,6051,571],{"className":6052},[570],[27,6054,6056],{"className":6055},[543],[27,6057,6059],{"className":6058,"style":971},[547],[27,6060],{},[27,6062,768],{"className":6063},[977,563],[27,6065,6067,6070],{"className":6066},[70,563],[27,6068,506],{"className":6069,"style":530},[70,71,563],[27,6071,6073],{"className":6072},[534],[27,6074,6076,6096],{"className":6075},[538,539],[27,6077,6079,6093],{"className":6078},[543],[27,6080,6082],{"className":6081,"style":946},[547],[27,6083,6084,6087],{"style":949},[27,6085],{"className":6086,"style":953},[555],[27,6088,6090],{"className":6089},[560,957,958,563],[27,6091,599],{"className":6092},[70,563],[27,6094,571],{"className":6095},[570],[27,6097,6099],{"className":6098},[543],[27,6100,6102],{"className":6101,"style":971},[547],[27,6103],{},[27,6105,768],{"className":6106},[977,563],[27,6108,6110,6113],{"className":6109},[70,563],[27,6111,506],{"className":6112,"style":530},[70,71,563],[27,6114,6116],{"className":6115},[534],[27,6117,6119,6139],{"className":6118},[538,539],[27,6120,6122,6136],{"className":6121},[543],[27,6123,6125],{"className":6124,"style":946},[547],[27,6126,6127,6130],{"style":949},[27,6128],{"className":6129,"style":953},[555],[27,6131,6133],{"className":6132},[560,957,958,563],[27,6134,670],{"className":6135},[70,563],[27,6137,571],{"className":6138},[570],[27,6140,6142],{"className":6141},[543],[27,6143,6145],{"className":6144,"style":971},[547],[27,6146],{},[27,6148,6149,6152],{"style":1063},[27,6150],{"className":6151,"style":921},[555],[27,6153],{"className":6154,"style":1071},[1070],[27,6156,6157,6160],{"style":1074},[27,6158],{"className":6159,"style":921},[555],[27,6161,6163],{"className":6162},[560,561,562,563],[27,6164,6166,6206,6246,6249,6289,6329,6332,6372],{"className":6165},[70,563],[27,6167,6169,6172],{"className":6168},[70,563],[27,6170,506],{"className":6171,"style":530},[70,71,563],[27,6173,6175],{"className":6174},[534],[27,6176,6178,6198],{"className":6177},[538,539],[27,6179,6181,6195],{"className":6180},[543],[27,6182,6184],{"className":6183,"style":946},[547],[27,6185,6186,6189],{"style":949},[27,6187],{"className":6188,"style":953},[555],[27,6190,6192],{"className":6191},[560,957,958,563],[27,6193,510],{"className":6194},[70,563],[27,6196,571],{"className":6197},[570],[27,6199,6201],{"className":6200},[543],[27,6202,6204],{"className":6203,"style":971},[547],[27,6205],{},[27,6207,6209,6212],{"className":6208},[70,563],[27,6210,742],{"className":6211,"style":838},[70,71,563],[27,6213,6215],{"className":6214},[534],[27,6216,6218,6238],{"className":6217},[538,539],[27,6219,6221,6235],{"className":6220},[543],[27,6222,6224],{"className":6223,"style":1142},[547],[27,6225,6226,6229],{"style":1145},[27,6227],{"className":6228,"style":953},[555],[27,6230,6232],{"className":6231},[560,957,958,563],[27,6233,149],{"className":6234},[70,71,563],[27,6236,571],{"className":6237},[570],[27,6239,6241],{"className":6240},[543],[27,6242,6244],{"className":6243,"style":1164},[547],[27,6245]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118.4550°",[27,7038,7040,7058],{"className":7039,"ariaHidden":57},[56],[27,7041,7043,7046,7049,7052,7055],{"className":7042},[61],[27,7044],{"className":7045,"style":2213},[65],[27,7047,2338],{"className":7048,"style":2354},[70,71],[27,7050],{"className":7051,"style":879},[878],[27,7053,748],{"className":7054},[883],[27,7056],{"className":7057,"style":879},[878],[27,7059,7061,7064],{"className":7060},[61],[27,7062],{"className":7063,"style":2313},[65],[27,7065,7067],{"className":7066},[70],"118.4550°",[27,7069,7071,7090],{"className":7070},[30],[27,7072,7074],{"className":7073},[34],[36,7075,7076],{"xmlns":38},[40,7077,7078,7088],{},[43,7079,7080,7082,7084,7086],{},[46,7081,2370],{"mathvariant":2299},[46,7083,149],{},[46,7085,48],{},[46,7087,179],{},[50,7089,2379],{"encoding":52},[27,7091,7093],{"className":7092,"ariaHidden":57},[56],[27,7094,7096,7099,7102,7105,7108],{"className":7095},[61],[27,7097],{"className":7098,"style":2389},[65],[27,7100,2370],{"className":7101},[70],[27,7103,149],{"className":7104},[70,71],[27,7106,48],{"className":7107,"style":72},[70,71],[27,7109,179],{"className":7110,"style":194},[70,71],", opposite ",[27,7113,7115,7128],{"className":7114},[30],[27,7116,7118],{"className":7117},[34],[36,7119,7120],{"xmlns":38},[40,7121,7122,7126],{},[43,7123,7124],{},[46,7125,210],{},[50,7127,210],{"encoding":52},[27,7129,7131],{"className":7130,"ariaHidden":57},[56],[27,7132,7134,7137],{"className":7133},[61],[27,7135],{"className":7136,"style":66},[65],[27,7138,210],{"className":7139,"style":225},[70,71],[2322,7141,7142,4907,7197,7111,7240,2460],{},[27,7143,7145,7166],{"className":7144},[30],[27,7146,7148],{"className":7147},[34],[36,7149,7150],{"xmlns":38},[40,7151,7152,7163],{},[43,7153,7154,7156,7158,7161],{},[46,7155,2477],{},[746,7157,748],{},[508,7159,7160],{},"104.3820",[46,7162,2300],{"mathvariant":2299},[50,7164,7165],{"encoding":52},"\\beta = 104.3820°",[27,7167,7169,7187],{"className":7168,"ariaHidden":57},[56],[27,7170,7172,7175,7178,7181,7184],{"className":7171},[61],[27,7173],{"className":7174,"style":523},[65],[27,7176,2477],{"className":7177,"style":2493},[70,71],[27,7179],{"className":7180,"style":879},[878],[27,7182,748],{"className":7183},[883],[27,7185],{"className":7186,"style":879},[878],[27,7188,7190,7193],{"className":7189},[61],[27,7191],{"className":7192,"style":2313},[65],[27,7194,7196],{"className":7195},[70],"104.3820°",[27,7198,7200,7219],{"className":7199},[30],[27,7201,7203],{"className":7202},[34],[36,7204,7205],{"xmlns":38},[40,7206,7207,7217],{},[43,7208,7209,7211,7213,7215],{},[46,7210,2370],{"mathvariant":2299},[46,7212,179],{},[46,7214,48],{},[46,7216,210],{},[50,7218,2516],{"encoding":52},[27,7220,7222],{"className":7221,"ariaHidden":57},[56],[27,7223,7225,7228,7231,7234,7237],{"className":7224},[61],[27,7226],{"className":7227,"style":2389},[65],[27,7229,2370],{"className":7230},[70],[27,7232,179],{"className":7233,"style":194},[70,71],[27,7235,48],{"className":7236,"style":72},[70,71],[27,7238,210],{"className":7239,"style":225},[70,71],[27,7241,7243,7256],{"className":7242},[30],[27,7244,7246],{"className":7245},[34],[36,7247,7248],{"xmlns":38},[40,7249,7250,7254],{},[43,7251,7252],{},[46,7253,149],{},[50,7255,149],{"encoding":52},[27,7257,7259],{"className":7258,"ariaHidden":57},[56],[27,7260,7262,7265],{"className":7261},[61],[27,7263],{"className":7264,"style":66},[65],[27,7266,149],{"className":7267},[70,71],[2322,7269,7270,4907,7325,7111,7368,2460],{},[27,7271,7273,7294],{"className":7272},[30],[27,7274,7276],{"className":7275},[34],[36,7277,7278],{"xmlns":38},[40,7279,7280,7291],{},[43,7281,7282,7284,7286,7289],{},[46,7283,2610],{},[746,7285,748],{},[508,7287,7288],{},"137.1630",[46,7290,2300],{"mathvariant":2299},[50,7292,7293],{"encoding":52},"\\gamma = 137.1630°",[27,7295,7297,7315],{"className":7296,"ariaHidden":57},[56],[27,7298,7300,7303,7306,7309,7312],{"className":7299},[61],[27,7301],{"className":7302,"style":2244},[65],[27,7304,2610],{"className":7305,"style":2626},[70,71],[27,7307],{"className":7308,"style":879},[878],[27,7310,748],{"className":7311},[883],[27,7313],{"className":7314,"style":879},[878],[27,7316,7318,7321],{"className":7317},[61],[27,7319],{"className":7320,"style":2313},[65],[27,7322,7324],{"className":7323},[70],"137.1630°",[27,7326,7328,7347],{"className":7327},[30],[27,7329,7331],{"className":7330},[34],[36,7332,7333],{"xmlns":38},[40,7334,7335,7345],{},[43,7336,7337,7339,7341,7343],{},[46,7338,2370],{"mathvariant":2299},[46,7340,210],{},[46,7342,48],{},[46,7344,149],{},[50,7346,2649],{"encoding":52},[27,7348,7350],{"className":7349,"ariaHidden":57},[56],[27,7351,7353,7356,7359,7362,7365],{"className":7352},[61],[27,7354],{"className":7355,"style":2389},[65],[27,7357,2370],{"className":7358},[70],[27,7360,210],{"className":7361,"style":225},[70,71],[27,7363,48],{"className":7364,"style":72},[70,71],[27,7366,149],{"className":7367},[70,71],[27,7369,7371,7384],{"className":7370},[30],[27,7372,7374],{"className":7373},[34],[36,7375,7376],{"xmlns":38},[40,7377,7378,7382],{},[43,7379,7380],{},[46,7381,179],{},[50,7383,179],{"encoding":52},[27,7385,7387],{"className":7386,"ariaHidden":57},[56],[27,7388,7390,7393],{"className":7389},[61],[27,7391],{"className":7392,"style":66},[65],[27,7394,179],{"className":7395,"style":194},[70,71],[15,7397,7398,7399,485],{},"Check: ",[27,7400,7402,7431],{"className":7401},[30],[27,7403,7405],{"className":7404},[34],[36,7406,7407],{"xmlns":38},[40,7408,7409,7428],{},[43,7410,7411,7413,7415,7417,7419,7421,7423,7426],{},[508,7412,7031],{},[746,7414,768],{},[508,7416,7160],{},[746,7418,768],{},[508,7420,7288],{},[746,7422,748],{},[508,7424,7425],{},"360.0000",[46,7427,2300],{"mathvariant":2299},[50,7429,7430],{"encoding":52},"118.4550 + 104.3820 + 137.1630 = 360.0000°",[27,7432,7434,7453,7471,7489],{"className":7433,"ariaHidden":57},[56],[27,7435,7437,7441,7444,7447,7450],{"className":7436},[61],[27,7438],{"className":7439,"style":7440},[65],"height:0.7278em;vertical-align:-0.0833em;",[27,7442,7031],{"className":7443},[70],[27,7445],{"className":7446,"style":2783},[878],[27,7448,768],{"className":7449},[977],[27,7451],{"className":7452,"style":2783},[878],[27,7454,7456,7459,7462,7465,7468],{"className":7455},[61],[27,7457],{"className":7458,"style":7440},[65],[27,7460,7160],{"className":7461},[70],[27,7463],{"className":7464,"style":2783},[878],[27,7466,768],{"className":7467},[977],[27,7469],{"className":7470,"style":2783},[878],[27,7472,7474,7477,7480,7483,7486],{"className":7473},[61],[27,7475],{"className":7476,"style":3214},[65],[27,7478,7288],{"className":7479},[70],[27,7481],{"className":7482,"style":879},[878],[27,7484,748],{"className":7485},[883],[27,7487],{"className":7488,"style":879},[878],[27,7490,7492,7495],{"className":7491},[61],[27,7493],{"className":7494,"style":2313},[65],[27,7496,7498],{"className":7497},[70],"360.0000°",[88,7500,7502,7503],{"id":7501},"interior-angles-of-abctriangle-abcabc","Interior angles of ",[27,7504,7506,7525],{"className":7505},[30],[27,7507,7509],{"className":7508},[34],[36,7510,7511],{"xmlns":38},[40,7512,7513,7523],{},[43,7514,7515,7517,7519,7521],{},[46,7516,2893],{"mathvariant":2299},[46,7518,149],{},[46,7520,179],{},[46,7522,210],{},[50,7524,2902],{"encoding":52},[27,7526,7528],{"className":7527,"ariaHidden":57},[56],[27,7529,7531,7534,7537,7540,7543],{"className":7530},[61],[27,7532],{"className":7533,"style":523},[65],[27,7535,2893],{"className":7536},[70],[27,7538,149],{"className":7539},[70,71],[27,7541,179],{"className":7542,"style":194},[70,71],[27,7544,210],{"className":7545,"style":225},[70,71],[2319,7547,7548,7818,8085],{},[2322,7549,7550,164,7656,7762,7763],{},[27,7551,7553,7582],{"className":7552},[30],[27,7554,7556],{"className":7555},[34],[36,7557,7558],{"xmlns":38},[40,7559,7560,7579],{},[43,7561,7562,7572,7574,7577],{},[502,7563,7564,7566],{},[46,7565,3664],{},[43,7567,7568,7570],{},[46,7569,149],{},[46,7571,179],{},[746,7573,748],{},[508,7575,7576],{},"90.0000",[46,7578,2300],{"mathvariant":2299},[50,7580,7581],{"encoding":52},"\\theta_{AB} = 90.0000°",[27,7583,7585,7646],{"className":7584,"ariaHidden":57},[56],[27,7586,7588,7591,7637,7640,7643],{"className":7587},[61],[27,7589],{"className":7590,"style":3703},[65],[27,7592,7594,7597],{"className":7593},[70],[27,7595,3664],{"className":7596,"style":3736},[70,71],[27,7598,7600],{"className":7599},[534],[27,7601,7603,7629],{"className":7602},[538,539],[27,7604,7606,7626],{"className":7605},[543],[27,7607,7609],{"className":7608,"style":851},[547],[27,7610,7611,7614],{"style":4563},[27,7612],{"className":7613,"style":556},[555],[27,7615,7617],{"className":7616},[560,561,562,563],[27,7618,7620,7623],{"className":7619},[70,563],[27,7621,149],{"className":7622},[70,71,563],[27,7624,179],{"className":7625,"style":194},[70,71,563],[27,7627,571],{"className":7628},[570],[27,7630,7632],{"className":7631},[543],[27,7633,7635],{"className":7634,"style":578},[547],[27,7636],{},[27,7638],{"className":7639,"style":879},[878],[27,7641,748],{"className":7642},[883],[27,7644],{"className":7645,"style":879},[878],[27,7647,7649,7652],{"className":7648},[61],[27,7650],{"className":7651,"style":2313},[65],[27,7653,7655],{"className":7654},[70],"90.0000°",[27,7657,7659,7688],{"className":7658},[30],[27,7660,7662],{"className":7661},[34],[36,7663,7664],{"xmlns":38},[40,7665,7666,7685],{},[43,7667,7668,7678,7680,7683],{},[502,7669,7670,7672],{},[46,7671,3664],{},[43,7673,7674,7676],{},[46,7675,149],{},[46,7677,210],{},[746,7679,748],{},[508,7681,7682],{},"33.6901",[46,7684,2300],{"mathvariant":2299},[50,7686,7687],{"encoding":52},"\\theta_{AC} = 33.6901°",[27,7689,7691,7752],{"className":7690,"ariaHidden":57},[56],[27,7692,7694,7697,7743,7746,7749],{"className":7693},[61],[27,7695],{"className":7696,"style":3703},[65],[27,7698,7700,7703],{"className":7699},[70],[27,7701,3664],{"className":7702,"style":3736},[70,71],[27,7704,7706],{"className":7705},[534],[27,7707,7709,7735],{"className":7708},[538,539],[27,7710,7712,7732],{"className":7711},[543],[27,7713,7715],{"className":7714,"style":851},[547],[27,7716,7717,7720],{"style":4563},[27,7718],{"className":7719,"style":556},[555],[27,7721,7723],{"className":7722},[560,561,562,563],[27,7724,7726,7729],{"className":7725},[70,563],[27,7727,149],{"className":7728},[70,71,563],[27,7730,210],{"className":7731,"style":225},[70,71,563],[27,7733,571],{"className":7734},[570],[27,7736,7738],{"className":7737},[543],[27,7739,7741],{"className":7740,"style":578},[547],[27,7742],{},[27,7744],{"className":7745,"style":879},[878],[27,7747,748],{"className":7748},[883],[27,7750],{"className":7751,"style":879},[878],[27,7753,7755,7758],{"className":7754},[61],[27,7756],{"className":7757,"style":2313},[65],[27,7759,7761],{"className":7760},[70],"33.6901°"," → ",[27,7764,7766,7787],{"className":7765},[30],[27,7767,7769],{"className":7768},[34],[36,7770,7771],{"xmlns":38},[40,7772,7773,7784],{},[43,7774,7775,7777,7779,7782],{},[46,7776,149],{},[746,7778,748],{},[508,7780,7781],{},"56.3099",[46,7783,2300],{"mathvariant":2299},[50,7785,7786],{"encoding":52},"A = 56.3099°",[27,7788,7790,7808],{"className":7789,"ariaHidden":57},[56],[27,7791,7793,7796,7799,7802,7805],{"className":7792},[61],[27,7794],{"className":7795,"style":66},[65],[27,7797,149],{"className":7798},[70,71],[27,7800],{"className":7801,"style":879},[878],[27,7803,748],{"className":7804},[883],[27,7806],{"className":7807,"style":879},[878],[27,7809,7811,7814],{"className":7810},[61],[27,7812],{"className":7813,"style":2313},[65],[27,7815,7817],{"className":7816},[70],"56.3099°",[2322,7819,7820,164,7926,7762,8032],{},[27,7821,7823,7852],{"className":7822},[30],[27,7824,7826],{"className":7825},[34],[36,7827,7828],{"xmlns":38},[40,7829,7830,7849],{},[43,7831,7832,7842,7844,7847],{},[502,7833,7834,7836],{},[46,7835,3664],{},[43,7837,7838,7840],{},[46,7839,179],{},[46,7841,210],{},[746,7843,748],{},[508,7845,7846],{},"326.3099",[46,7848,2300],{"mathvariant":2299},[50,7850,7851],{"encoding":52},"\\theta_{BC} = 326.3099°",[27,7853,7855,7916],{"className":7854,"ariaHidden":57},[56],[27,7856,7858,7861,7907,7910,7913],{"className":7857},[61],[27,7859],{"className":7860,"style":3703},[65],[27,7862,7864,7867],{"className":7863},[70],[27,7865,3664],{"className":7866,"style":3736},[70,71],[27,7868,7870],{"className":7869},[534],[27,7871,7873,7899],{"className":7872},[538,539],[27,7874,7876,7896],{"className":7875},[543],[27,7877,7879],{"className":7878,"style":851},[547],[27,7880,7881,7884],{"style":4563},[27,7882],{"className":7883,"style":556},[555],[27,7885,7887],{"className":7886},[560,561,562,563],[27,7888,7890,7893],{"className":7889},[70,563],[27,7891,179],{"className":7892,"style":194},[70,71,563],[27,7894,210],{"className":7895,"style":225},[70,71,563],[27,7897,571],{"className":7898},[570],[27,7900,7902],{"className":7901},[543],[27,7903,7905],{"className":7904,"style":578},[547],[27,7906],{},[27,7908],{"className":7909,"style":879},[878],[27,7911,748],{"className":7912},[883],[27,7914],{"className":7915,"style":879},[878],[27,7917,7919,7922],{"className":7918},[61],[27,7920],{"className":7921,"style":2313},[65],[27,7923,7925],{"className":7924},[70],"326.3099°",[27,7927,7929,7958],{"className":7928},[30],[27,7930,7932],{"className":7931},[34],[36,7933,7934],{"xmlns":38},[40,7935,7936,7955],{},[43,7937,7938,7948,7950,7953],{},[502,7939,7940,7942],{},[46,7941,3664],{},[43,7943,7944,7946],{},[46,7945,179],{},[46,7947,149],{},[746,7949,748],{},[508,7951,7952],{},"270.0000",[46,7954,2300],{"mathvariant":2299},[50,7956,7957],{"encoding":52},"\\theta_{BA} = 270.0000°",[27,7959,7961,8022],{"className":7960,"ariaHidden":57},[56],[27,7962,7964,7967,8013,8016,8019],{"className":7963},[61],[27,7965],{"className":7966,"style":3703},[65],[27,7968,7970,7973],{"className":7969},[70],[27,7971,3664],{"className":7972,"style":3736},[70,71],[27,7974,7976],{"className":7975},[534],[27,7977,7979,8005],{"className":7978},[538,539],[27,7980,7982,8002],{"className":7981},[543],[27,7983,7985],{"className":7984,"style":851},[547],[27,7986,7987,7990],{"style":4563},[27,7988],{"className":7989,"style":556},[555],[27,7991,7993],{"className":7992},[560,561,562,563],[27,7994,7996,7999],{"className":7995},[70,563],[27,7997,179],{"className":7998,"style":194},[70,71,563],[27,8000,149],{"className":8001},[70,71,563],[27,8003,571],{"className":8004},[570],[27,8006,8008],{"className":8007},[543],[27,8009,8011],{"className":8010,"style":578},[547],[27,8012],{},[27,8014],{"className":8015,"style":879},[878],[27,8017,748],{"className":8018},[883],[27,8020],{"className":8021,"style":879},[878],[27,8023,8025,8028],{"className":8024},[61],[27,8026],{"className":8027,"style":2313},[65],[27,8029,8031],{"className":8030},[70],"270.0000°",[27,8033,8035,8055],{"className":8034},[30],[27,8036,8038],{"className":8037},[34],[36,8039,8040],{"xmlns":38},[40,8041,8042,8052],{},[43,8043,8044,8046,8048,8050],{},[46,8045,179],{},[746,8047,748],{},[508,8049,7781],{},[46,8051,2300],{"mathvariant":2299},[50,8053,8054],{"encoding":52},"B = 56.3099°",[27,8056,8058,8076],{"className":8057,"ariaHidden":57},[56],[27,8059,8061,8064,8067,8070,8073],{"className":8060},[61],[27,8062],{"className":8063,"style":66},[65],[27,8065,179],{"className":8066,"style":194},[70,71],[27,8068],{"className":8069,"style":879},[878],[27,8071,748],{"className":8072},[883],[27,8074],{"className":8075,"style":879},[878],[27,8077,8079,8082],{"className":8078},[61],[27,8080],{"className":8081,"style":2313},[65],[27,8083,7817],{"className":8084},[70],[2322,8086,8087,164,8193,7762,8299],{},[27,8088,8090,8119],{"className":8089},[30],[27,8091,8093],{"className":8092},[34],[36,8094,8095],{"xmlns":38},[40,8096,8097,8116],{},[43,8098,8099,8109,8111,8114],{},[502,8100,8101,8103],{},[46,8102,3664],{},[43,8104,8105,8107],{},[46,8106,210],{},[46,8108,149],{},[746,8110,748],{},[508,8112,8113],{},"213.6901",[46,8115,2300],{"mathvariant":2299},[50,8117,8118],{"encoding":52},"\\theta_{CA} = 213.6901°",[27,8120,8122,8183],{"className":8121,"ariaHidden":57},[56],[27,8123,8125,8128,8174,8177,8180],{"className":8124},[61],[27,8126],{"className":8127,"style":3703},[65],[27,8129,8131,8134],{"className":8130},[70],[27,8132,3664],{"className":8133,"style":3736},[70,71],[27,8135,8137],{"className":8136},[534],[27,8138,8140,8166],{"className":8139},[538,539],[27,8141,8143,8163],{"className":8142},[543],[27,8144,8146],{"className":8145,"style":851},[547],[27,8147,8148,8151],{"style":4563},[27,8149],{"className":8150,"style":556},[555],[27,8152,8154],{"className":8153},[560,561,562,563],[27,8155,8157,8160],{"className":8156},[70,563],[27,8158,210],{"className":8159,"style":225},[70,71,563],[27,8161,149],{"className":8162},[70,71,563],[27,8164,571],{"className":8165},[570],[27,8167,8169],{"className":8168},[543],[27,8170,8172],{"className":8171,"style":578},[547],[27,8173],{},[27,8175],{"className":8176,"style":879},[878],[27,8178,748],{"className":8179},[883],[27,8181],{"className":8182,"style":879},[878],[27,8184,8186,8189],{"className":8185},[61],[27,8187],{"className":8188,"style":2313},[65],[27,8190,8192],{"className":8191},[70],"213.6901°",[27,8194,8196,8225],{"className":8195},[30],[27,8197,8199],{"className":8198},[34],[36,8200,8201],{"xmlns":38},[40,8202,8203,8222],{},[43,8204,8205,8215,8217,8220],{},[502,8206,8207,8209],{},[46,8208,3664],{},[43,8210,8211,8213],{},[46,8212,210],{},[46,8214,179],{},[746,8216,748],{},[508,8218,8219],{},"146.3099",[46,8221,2300],{"mathvariant":2299},[50,8223,8224],{"encoding":52},"\\theta_{CB} = 146.3099°",[27,8226,8228,8289],{"className":8227,"ariaHidden":57},[56],[27,8229,8231,8234,8280,8283,8286],{"className":8230},[61],[27,8232],{"className":8233,"style":3703},[65],[27,8235,8237,8240],{"className":8236},[70],[27,8238,3664],{"className":8239,"style":3736},[70,71],[27,8241,8243],{"className":8242},[534],[27,8244,8246,8272],{"className":8245},[538,539],[27,8247,8249,8269],{"className":8248},[543],[27,8250,8252],{"className":8251,"style":851},[547],[27,8253,8254,8257],{"style":4563},[27,8255],{"className":8256,"style":556},[555],[27,8258,8260],{"className":8259},[560,561,562,563],[27,8261,8263,8266],{"className":8262},[70,563],[27,8264,210],{"className":8265,"style":225},[70,71,563],[27,8267,179],{"className":8268,"style":194},[70,71,563],[27,8270,571],{"className":8271},[570],[27,8273,8275],{"className":8274},[543],[27,8276,8278],{"className":8277,"style":578},[547],[27,8279],{},[27,8281],{"className":8282,"style":879},[878],[27,8284,748],{"className":8285},[883],[27,8287],{"className":8288,"style":879},[878],[27,8290,8292,8295],{"className":8291},[61],[27,8293],{"className":8294,"style":2313},[65],[27,8296,8298],{"className":8297},[70],"146.3099°",[27,8300,8302,8323],{"className":8301},[30],[27,8303,8305],{"className":8304},[34],[36,8306,8307],{"xmlns":38},[40,8308,8309,8320],{},[43,8310,8311,8313,8315,8318],{},[46,8312,210],{},[746,8314,748],{},[508,8316,8317],{},"67.3802",[46,8319,2300],{"mathvariant":2299},[50,8321,8322],{"encoding":52},"C = 67.3802°",[27,8324,8326,8344],{"className":8325,"ariaHidden":57},[56],[27,8327,8329,8332,8335,8338,8341],{"className":8328},[61],[27,8330],{"className":8331,"style":66},[65],[27,8333,210],{"className":8334,"style":225},[70,71],[27,8336],{"className":8337,"style":879},[878],[27,8339,748],{"className":8340},[883],[27,8342],{"className":8343,"style":879},[878],[27,8345,8347,8350],{"className":8346},[61],[27,8348],{"className":8349,"style":2313},[65],[27,8351,8353],{"className":8352},[70],"67.3802°",[15,8355,8356,8357,485],{},"Sum: ",[27,8358,8360,8377],{"className":8359},[30],[27,8361,8363],{"className":8362},[34],[36,8364,8365],{"xmlns":38},[40,8366,8367,8374],{},[43,8368,8369,8372],{},[508,8370,8371],{},"180.0000",[46,8373,2300],{"mathvariant":2299},[50,8375,8376],{"encoding":52},"180.0000°",[27,8378,8380],{"className":8379,"ariaHidden":57},[56],[27,8381,8383,8386],{"className":8382},[61],[27,8384],{"className":8385,"style":2313},[65],[27,8387,8376],{"className":8388},[70],[88,8390,8392],{"id":8391},"cotangents-and-weights","Cotangents and weights",[15,8394,8395],{},[27,8396,8398,8447],{"className":8397},[30],[27,8399,8401],{"className":8400},[34],[36,8402,8403],{"xmlns":38},[40,8404,8405,8444],{},[43,8406,8407,8409,8411,8413,8415,8418,8421,8423,8425,8427,8429,8431,8433,8435,8437,8439,8441],{},[46,8408,2973],{},[746,8410,2976],{},[46,8412,149],{},[746,8414,3438],{},[508,8416,8417],{},"0.666667",[878,8419],{"width":8420},"1em",[46,8422,2973],{},[746,8424,2976],{},[46,8426,179],{},[746,8428,3438],{},[508,8430,8417],{},[878,8432],{"width":8420},[46,8434,2973],{},[746,8436,2976],{},[46,8438,210],{},[746,8440,3438],{},[508,8442,8443],{},"0.416667",[50,8445,8446],{"encoding":52},"\\cot A \\approx 0.666667 \\quad \\cot B \\approx 0.666667 \\quad \\cot C \\approx 0.416667",[27,8448,8450,8474,8508,8541],{"className":8449,"ariaHidden":57},[56],[27,8451,8453,8456,8459,8462,8465,8468,8471],{"className":8452},[61],[27,8454],{"className":8455,"style":66},[65],[27,8457,2973],{"className":8458},[3050],[27,8460],{"className":8461,"style":3054},[878],[27,8463,149],{"className":8464},[70,71],[27,8466],{"className":8467,"style":879},[878],[27,8469,3438],{"className":8470},[883],[27,8472],{"className":8473,"style":879},[878],[27,8475,8477,8480,8483,8487,8490,8493,8496,8499,8502,8505],{"className":8476},[61],[27,8478],{"className":8479,"style":66},[65],[27,8481,8417],{"className":8482},[70],[27,8484],{"className":8485,"style":8486},[878],"margin-right:1em;",[27,8488],{"className":8489,"style":3054},[878],[27,8491,2973],{"className":8492},[3050],[27,8494],{"className":8495,"style":3054},[878],[27,8497,179],{"className":8498,"style":194},[70,71],[27,8500],{"className":8501,"style":879},[878],[27,8503,3438],{"className":8504},[883],[27,8506],{"className":8507,"style":879},[878],[27,8509,8511,8514,8517,8520,8523,8526,8529,8532,8535,8538],{"className":8510},[61],[27,8512],{"className":8513,"style":66},[65],[27,8515,8417],{"className":8516},[70],[27,8518],{"className":8519,"style":8486},[878],[27,8521],{"className":8522,"style":3054},[878],[27,8524,2973],{"className":8525},[3050],[27,8527],{"className":8528,"style":3054},[878],[27,8530,210],{"className":8531,"style":225},[70,71],[27,8533],{"className":8534,"style":879},[878],[27,8536,3438],{"className":8537},[883],[27,8539],{"className":8540,"style":879},[878],[27,8542,8544,8547],{"className":8543},[61],[27,8545],{"className":8546,"style":3214},[65],[27,8548,8443],{"className":8549},[70],[15,8551,8552],{},[27,8553,8555,8610],{"className":8554},[30],[27,8556,8558],{"className":8557},[34],[36,8559,8560],{"xmlns":38},[40,8561,8562,8607],{},[43,8563,8564,8566,8568,8570,8572,8574,8577,8579,8581,8583,8585,8587,8589,8592,8594,8596,8598,8600,8602,8604],{},[46,8565,2973],{},[746,8567,2976],{},[46,8569,2338],{},[746,8571,3438],{},[746,8573,2981],{},[508,8575,8576],{},"0.541951",[878,8578],{"width":8420},[46,8580,2973],{},[746,8582,2976],{},[46,8584,2477],{},[746,8586,3438],{},[746,8588,2981],{},[508,8590,8591],{},"0.256428",[878,8593],{"width":8420},[46,8595,2973],{},[746,8597,2976],{},[46,8599,2610],{},[746,8601,3438],{},[746,8603,2981],{},[508,8605,8606],{},"1.078441",[50,8608,8609],{"encoding":52},"\\cot\\alpha \\approx -0.541951 \\quad \\cot\\beta \\approx -0.256428 \\quad \\cot\\gamma \\approx -1.078441",[27,8611,8613,8637,8673,8710],{"className":8612,"ariaHidden":57},[56],[27,8614,8616,8619,8622,8625,8628,8631,8634],{"className":8615},[61],[27,8617],{"className":8618,"style":3189},[65],[27,8620,2973],{"className":8621},[3050],[27,8623],{"className":8624,"style":3054},[878],[27,8626,2338],{"className":8627,"style":2354},[70,71],[27,8629],{"className":8630,"style":879},[878],[27,8632,3438],{"className":8633},[883],[27,8635],{"className":8636,"style":879},[878],[27,8638,8640,8643,8646,8649,8652,8655,8658,8661,8664,8667,8670],{"className":8639},[61],[27,8641],{"className":8642,"style":523},[65],[27,8644,2981],{"className":8645},[70],[27,8647,8576],{"className":8648},[70],[27,8650],{"className":8651,"style":8486},[878],[27,8653],{"className":8654,"style":3054},[878],[27,8656,2973],{"className":8657},[3050],[27,8659],{"className":8660,"style":3054},[878],[27,8662,2477],{"className":8663,"style":2493},[70,71],[27,8665],{"className":8666,"style":879},[878],[27,8668,3438],{"className":8669},[883],[27,8671],{"className":8672,"style":879},[878],[27,8674,8676,8680,8683,8686,8689,8692,8695,8698,8701,8704,8707],{"className":8675},[61],[27,8677],{"className":8678,"style":8679},[65],"height:0.8389em;vertical-align:-0.1944em;",[27,8681,2981],{"className":8682},[70],[27,8684,8591],{"className":8685},[70],[27,8687],{"className":8688,"style":8486},[878],[27,8690],{"className":8691,"style":3054},[878],[27,8693,2973],{"className":8694},[3050],[27,8696],{"className":8697,"style":3054},[878],[27,8699,2610],{"className":8700,"style":2626},[70,71],[27,8702],{"className":8703,"style":879},[878],[27,8705,3438],{"className":8706},[883],[27,8708],{"className":8709,"style":879},[878],[27,8711,8713,8716,8719],{"className":8712},[61],[27,8714],{"className":8715,"style":7440},[65],[27,8717,2981],{"className":8718},[70],[27,8720,8606],{"className":8721},[70],[15,8723,8724],{},[27,8725,8727,8796],{"className":8726},[30],[27,8728,8730],{"className":8729},[34],[36,8731,8732],{"xmlns":38},[40,8733,8734,8793],{},[43,8735,8736,8742,8744,8762,8764,8767,8769,8775,8777,8780,8782,8788,8790],{},[502,8737,8738,8740],{},[46,8739,506],{},[508,8741,510],{},[746,8743,748],{},[750,8745,8746,8748],{},[508,8747,510],{},[43,8749,8750,8752,8754,8756,8758,8760],{},[508,8751,8417],{},[746,8753,2981],{},[746,8755,3678],{"stretchy":3677},[746,8757,2981],{},[508,8759,8591],{},[746,8761,2460],{"stretchy":3677},[746,8763,3438],{},[508,8765,8766],{},"1.083",[878,8768],{"width":2995},[502,8770,8771,8773],{},[46,8772,506],{},[508,8774,599],{},[746,8776,3438],{},[508,8778,8779],{},"0.573",[878,8781],{"width":2995},[502,8783,8784,8786],{},[46,8785,506],{},[508,8787,670],{},[746,8789,3438],{},[508,8791,8792],{},"1.043",[50,8794,8795],{"encoding":52},"f_1 = \\frac{1}{0.666667 - (-0.256428)} \\approx 1.083 \\qquad f_2 \\approx 0.573 \\qquad f_3 \\approx 1.043",[27,8797,8799,8854,8954,9015,9076],{"className":8798,"ariaHidden":57},[56],[27,8800,8802,8805,8845,8848,8851],{"className":8801},[61],[27,8803],{"className":8804,"style":523},[65],[27,8806,8808,8811],{"className":8807},[70],[27,8809,506],{"className":8810,"style":530},[70,71],[27,8812,8814],{"className":8813},[534],[27,8815,8817,8837],{"className":8816},[538,539],[27,8818,8820,8834],{"className":8819},[543],[27,8821,8823],{"className":8822,"style":548},[547],[27,8824,8825,8828],{"style":551},[27,8826],{"className":8827,"style":556},[555],[27,8829,8831],{"className":8830},[560,561,562,563],[27,8832,510],{"className":8833},[70,563],[27,8835,571],{"className":8836},[570],[27,8838,8840],{"className":8839},[543],[27,8841,8843],{"className":8842,"style":578},[547],[27,8844],{},[27,8846],{"className":8847,"style":879},[878],[27,8849,748],{"className":8850},[883],[27,8852],{"className":8853,"style":879},[878],[27,8855,8857,8861,8945,8948,8951],{"className":8856},[61],[27,8858],{"className":8859,"style":8860},[65],"height:1.3651em;vertical-align:-0.52em;",[27,8862,8864,8867,8942],{"className":8863},[70],[27,8865],{"className":8866},[900,901],[27,8868,8870],{"className":8869},[750],[27,8871,8873,8933],{"className":8872},[538,539],[27,8874,8876,8930],{"className":8875},[543],[27,8877,8879,8908,8916],{"className":8878,"style":5206},[547],[27,8880,8881,8884],{"style":917},[27,8882],{"className":8883,"style":921},[555],[27,8885,8887],{"className":8886},[560,561,562,563],[27,8888,8890,8893,8896,8899,8902,8905],{"className":8889},[70,563],[27,8891,8417],{"className":8892},[70,563],[27,8894,2981],{"className":8895},[977,563],[27,8897,3678],{"className":8898},[900,563],[27,8900,2981],{"className":8901},[70,563],[27,8903,8591],{"className":8904},[70,563],[27,8906,2460],{"className":8907},[1348,563],[27,8909,8910,8913],{"style":1063},[27,8911],{"className":8912,"style":921},[555],[27,8914],{"className":8915,"style":1071},[1070],[27,8917,8918,8921],{"style":5270},[27,8919],{"className":8920,"style":921},[555],[27,8922,8924],{"className":8923},[560,561,562,563],[27,8925,8927],{"className":8926},[70,563],[27,8928,510],{"className":8929},[70,563],[27,8931,571],{"className":8932},[570],[27,8934,8936],{"className":8935},[543],[27,8937,8940],{"className":8938,"style":8939},[547],"height:0.52em;",[27,8941],{},[27,8943],{"className":8944},[1348,901],[27,8946],{"className":8947,"style":879},[878],[27,8949,3438],{"className":8950},[883],[27,8952],{"className":8953,"style":879},[878],[27,8955,8957,8960,8963,8966,9006,9009,9012],{"className":8956},[61],[27,8958],{"className":8959,"style":523},[65],[27,8961,8766],{"className":8962},[70],[27,8964],{"className":8965,"style":3103},[878],[27,8967,8969,8972],{"className":8968},[70],[27,8970,506],{"className":8971,"style":530},[70,71],[27,8973,8975],{"className":8974},[534],[27,8976,8978,8998],{"className":8977},[538,539],[27,8979,8981,8995],{"className":8980},[543],[27,8982,8984],{"className":8983,"style":548},[547],[27,8985,8986,8989],{"style":551},[27,8987],{"className":8988,"style":556},[555],[27,8990,8992],{"className":8991},[560,561,562,563],[27,8993,599],{"className":8994},[70,563],[27,8996,571],{"className":8997},[570],[27,8999,9001],{"className":9000},[543],[27,9002,9004],{"className":9003,"style":578},[547],[27,9005],{},[27,9007],{"className":9008,"style":879},[878],[27,9010,3438],{"className":9011},[883],[27,9013],{"className":9014,"style":879},[878],[27,9016,9018,9021,9024,9027,9067,9070,9073],{"className":9017},[61],[27,9019],{"className":9020,"style":523},[65],[27,9022,8779],{"className":9023},[70],[27,9025],{"className":9026,"style":3103},[878],[27,9028,9030,9033],{"className":9029},[70],[27,9031,506],{"className":9032,"style":530},[70,71],[27,9034,9036],{"className":9035},[534],[27,9037,9039,9059],{"className":9038},[538,539],[27,9040,9042,9056],{"className":9041},[543],[27,9043,9045],{"className":9044,"style":548},[547],[27,9046,9047,9050],{"style":551},[27,9048],{"className":9049,"style":556},[555],[27,9051,9053],{"className":9052},[560,561,562,563],[27,9054,670],{"className":9055},[70,563],[27,9057,571],{"className":9058},[570],[27,9060,9062],{"className":9061},[543],[27,9063,9065],{"className":9064,"style":578},[547],[27,9066],{},[27,9068],{"className":9069,"style":879},[878],[27,9071,3438],{"className":9072},[883],[27,9074],{"className":9075,"style":879},[878],[27,9077,9079,9082],{"className":9078},[61],[27,9080],{"className":9081,"style":3214},[65],[27,9083,8792],{"className":9084},[70],[15,9086,9087],{},[27,9088,9090,9112],{"className":9089},[30],[27,9091,9093],{"className":9092},[34],[36,9094,9095],{"xmlns":38},[40,9096,9097,9109],{},[43,9098,9099,9102,9104,9106],{},[746,9100,9101],{},"∑",[46,9103,506],{},[746,9105,3438],{},[508,9107,9108],{},"2.699",[50,9110,9111],{"encoding":52},"\\sum f \\approx 2.699",[27,9113,9115,9142],{"className":9114,"ariaHidden":57},[56],[27,9116,9118,9121,9127,9130,9133,9136,9139],{"className":9117},[61],[27,9119],{"className":9120,"style":3811},[65],[27,9122,9101],{"className":9123,"style":9126},[3050,9124,9125],"op-symbol","small-op","position:relative;top:0em;",[27,9128],{"className":9129,"style":3054},[878],[27,9131,506],{"className":9132,"style":530},[70,71],[27,9134],{"className":9135,"style":879},[878],[27,9137,3438],{"className":9138},[883],[27,9140],{"className":9141,"style":879},[878],[27,9143,9145,9148],{"className":9144},[61],[27,9146],{"className":9147,"style":3214},[65],[27,9149,9108],{"className":9150},[70],[88,9152,9154],{"id":9153},"result","Result",[15,9156,9157],{},[27,9158,9160,9201],{"className":9159},[30],[27,9161,9163],{"className":9162},[34],[36,9164,9165],{"xmlns":38},[40,9166,9167,9198],{},[43,9168,9169,9175,9177,9180,9183,9185,9191,9193,9196],{},[502,9170,9171,9173],{},[46,9172,742],{},[46,9174,48],{},[746,9176,3438],{},[508,9178,9179],{},"1810.962",[3879,9181,9182],{}," m",[878,9184],{"width":2995},[502,9186,9187,9189],{},[46,9188,1367],{},[46,9190,48],{},[746,9192,3438],{},[508,9194,9195],{},"2579.650",[3879,9197,9182],{},[50,9199,9200],{"encoding":52},"E_P \\approx 1810.962 \\text{ m} \\qquad N_P \\approx 2579.650 \\text{ m}",[27,9202,9204,9259,9326],{"className":9203,"ariaHidden":57},[56],[27,9205,9207,9210,9250,9253,9256],{"className":9206},[61],[27,9208],{"className":9209,"style":831},[65],[27,9211,9213,9216],{"className":9212},[70],[27,9214,742],{"className":9215,"style":838},[70,71],[27,9217,9219],{"className":9218},[534],[27,9220,9222,9242],{"className":9221},[538,539],[27,9223,9225,9239],{"className":9224},[543],[27,9226,9228],{"className":9227,"style":851},[547],[27,9229,9230,9233],{"style":854},[27,9231],{"className":9232,"style":556},[555],[27,9234,9236],{"className":9235},[560,561,562,563],[27,9237,48],{"className":9238,"style":72},[70,71,563],[27,9240,571],{"className":9241},[570],[27,9243,9245],{"className":9244},[543],[27,9246,9248],{"className":9247,"style":578},[547],[27,9249],{},[27,9251],{"className":9252,"style":879},[878],[27,9254,3438],{"className":9255},[883],[27,9257],{"className":9258,"style":879},[878],[27,9260,9262,9265,9268,9274,9277,9317,9320,9323],{"className":9261},[61],[27,9263],{"className":9264,"style":831},[65],[27,9266,9179],{"className":9267},[70],[27,9269,9271],{"className":9270},[70,3486],[27,9272,9182],{"className":9273},[70],[27,9275],{"className":9276,"style":3103},[878],[27,9278,9280,9283],{"className":9279},[70],[27,9281,1367],{"className":9282,"style":1458},[70,71],[27,9284,9286],{"className":9285},[534],[27,9287,9289,9309],{"className":9288},[538,539],[27,9290,9292,9306],{"className":9291},[543],[27,9293,9295],{"className":9294,"style":851},[547],[27,9296,9297,9300],{"style":1473},[27,9298],{"className":9299,"style":556},[555],[27,9301,9303],{"className":9302},[560,561,562,563],[27,9304,48],{"className":9305,"style":72},[70,71,563],[27,9307,571],{"className":9308},[570],[27,9310,9312],{"className":9311},[543],[27,9313,9315],{"className":9314,"style":578},[547],[27,9316],{},[27,9318],{"className":9319,"style":879},[878],[27,9321,3438],{"className":9322},[883],[27,9324],{"className":9325,"style":879},[878],[27,9327,9329,9332,9335],{"className":9328},[61],[27,9330],{"className":9331,"style":3214},[65],[27,9333,9195],{"className":9334},[70],[27,9336,9338],{"className":9337},[70,3486],[27,9339,9182],{"className":9340},[70],[15,9342,9343,9344,9346],{},"Enter the same values in the ",[75,9345,78],{"href":77}," to verify.",[10,9348,9350],{"id":9349},"checking-your-result","Checking Your Result",[88,9352,9354],{"id":9353},"back-bearing-check","Back-bearing check",[15,9356,9357,9358,9386,9387,9459,9460,164,9488,164,9516,9544],{},"From computed ",[27,9359,9361,9374],{"className":9360},[30],[27,9362,9364],{"className":9363},[34],[36,9365,9366],{"xmlns":38},[40,9367,9368,9372],{},[43,9369,9370],{},[46,9371,48],{},[50,9373,48],{"encoding":52},[27,9375,9377],{"className":9376,"ariaHidden":57},[56],[27,9378,9380,9383],{"className":9379},[61],[27,9381],{"className":9382,"style":66},[65],[27,9384,48],{"className":9385,"style":72},[70,71],", calculate grid bearings to each control point using ",[27,9388,9390,9420],{"className":9389},[30],[27,9391,9393],{"className":9392},[34],[36,9394,9395],{"xmlns":38},[40,9396,9397,9417],{},[43,9398,9399,9402,9404,9407,9409,9411,9413,9415],{},[3879,9400,9401],{},"atan2",[746,9403,3678],{"stretchy":3677},[46,9405,9406],{"mathvariant":2299},"Δ",[46,9408,742],{},[746,9410,3907],{"separator":57},[46,9412,9406],{"mathvariant":2299},[46,9414,1367],{},[746,9416,2460],{"stretchy":3677},[50,9418,9419],{"encoding":52},"\\text{atan2}(\\Delta E, \\Delta N)",[27,9421,9423],{"className":9422,"ariaHidden":57},[56],[27,9424,9426,9429,9435,9438,9441,9444,9447,9450,9453,9456],{"className":9425},[61],[27,9427],{"className":9428,"style":3811},[65],[27,9430,9432],{"className":9431},[70,3486],[27,9433,9401],{"className":9434},[70],[27,9436,3678],{"className":9437},[900],[27,9439,9406],{"className":9440},[70],[27,9442,742],{"className":9443,"style":838},[70,71],[27,9445,3907],{"className":9446},[4132],[27,9448],{"className":9449,"style":3054},[878],[27,9451,9406],{"className":9452},[70],[27,9454,1367],{"className":9455,"style":1458},[70,71],[27,9457,2460],{"className":9458},[1348],", then reconstruct ",[27,9461,9463,9476],{"className":9462},[30],[27,9464,9466],{"className":9465},[34],[36,9467,9468],{"xmlns":38},[40,9469,9470,9474],{},[43,9471,9472],{},[46,9473,2338],{},[50,9475,2341],{"encoding":52},[27,9477,9479],{"className":9478,"ariaHidden":57},[56],[27,9480,9482,9485],{"className":9481},[61],[27,9483],{"className":9484,"style":2213},[65],[27,9486,2338],{"className":9487,"style":2354},[70,71],[27,9489,9491,9504],{"className":9490},[30],[27,9492,9494],{"className":9493},[34],[36,9495,9496],{"xmlns":38},[40,9497,9498,9502],{},[43,9499,9500],{},[46,9501,2477],{},[50,9503,2480],{"encoding":52},[27,9505,9507],{"className":9506,"ariaHidden":57},[56],[27,9508,9510,9513],{"className":9509},[61],[27,9511],{"className":9512,"style":523},[65],[27,9514,2477],{"className":9515,"style":2493},[70,71],[27,9517,9519,9532],{"className":9518},[30],[27,9520,9522],{"className":9521},[34],[36,9523,9524],{"xmlns":38},[40,9525,9526,9530],{},[43,9527,9528],{},[46,9529,2610],{},[50,9531,2613],{"encoding":52},[27,9533,9535],{"className":9534,"ariaHidden":57},[56],[27,9536,9538,9541],{"className":9537},[61],[27,9539],{"className":9540,"style":2244},[65],[27,9542,2610],{"className":9543,"style":2626},[70,71],". Reconstructed angles should match field values within a few seconds of arc.",[88,9546,9548],{"id":9547},"alternative-methods","Alternative methods",[2319,9550,9551,9557,9674],{},[2322,9552,9553,9556],{},[22,9554,9555],{},"Collins point"," — auxiliary construction on the circumcircle",[2322,9558,9559,9562,9563,4745,9619],{},[22,9560,9561],{},"Cassini"," — intersecting auxiliary circles through ",[27,9564,9566,9588],{"className":9565},[30],[27,9567,9569],{"className":9568},[34],[36,9570,9571],{"xmlns":38},[40,9572,9573,9585],{},[43,9574,9575,9577,9579,9581,9583],{},[46,9576,149],{},[746,9578,3907],{"separator":57},[46,9580,179],{},[746,9582,3907],{"separator":57},[46,9584,48],{},[50,9586,9587],{"encoding":52},"A,B,P",[27,9589,9591],{"className":9590,"ariaHidden":57},[56],[27,9592,9594,9598,9601,9604,9607,9610,9613,9616],{"className":9593},[61],[27,9595],{"className":9596,"style":9597},[65],"height:0.8778em;vertical-align:-0.1944em;",[27,9599,149],{"className":9600},[70,71],[27,9602,3907],{"className":9603},[4132],[27,9605],{"className":9606,"style":3054},[878],[27,9608,179],{"className":9609,"style":194},[70,71],[27,9611,3907],{"className":9612},[4132],[27,9614],{"className":9615,"style":3054},[878],[27,9617,48],{"className":9618,"style":72},[70,71],[27,9620,9622,9644],{"className":9621},[30],[27,9623,9625],{"className":9624},[34],[36,9626,9627],{"xmlns":38},[40,9628,9629,9641],{},[43,9630,9631,9633,9635,9637,9639],{},[46,9632,179],{},[746,9634,3907],{"separator":57},[46,9636,210],{},[746,9638,3907],{"separator":57},[46,9640,48],{},[50,9642,9643],{"encoding":52},"B,C,P",[27,9645,9647],{"className":9646,"ariaHidden":57},[56],[27,9648,9650,9653,9656,9659,9662,9665,9668,9671],{"className":9649},[61],[27,9651],{"className":9652,"style":9597},[65],[27,9654,179],{"className":9655,"style":194},[70,71],[27,9657,3907],{"className":9658},[4132],[27,9660],{"className":9661,"style":3054},[878],[27,9663,210],{"className":9664,"style":225},[70,71],[27,9666,3907],{"className":9667},[4132],[27,9669],{"className":9670,"style":3054},[878],[27,9672,48],{"className":9673,"style":72},[70,71],[2322,9675,9676,9679],{},[22,9677,9678],{},"Least-squares resection"," — when distances and redundant observations are available",[10,9681,9683],{"id":9682},"limitations-and-practical-warnings","Limitations and Practical Warnings",[88,9685,9687],{"id":9686},"sensitivity-to-angular-errors","Sensitivity to angular errors",[15,9689,9690,9691,4745,9721,9751,9752,9780],{},"Cotangents blow up near ",[27,9692,9694,9709],{"className":9693},[30],[27,9695,9697],{"className":9696},[34],[36,9698,9699],{"xmlns":38},[40,9700,9701,9707],{},[43,9702,9703,9705],{},[508,9704,2843],{},[46,9706,2300],{"mathvariant":2299},[50,9708,4672],{"encoding":52},[27,9710,9712],{"className":9711,"ariaHidden":57},[56],[27,9713,9715,9718],{"className":9714},[61],[27,9716],{"className":9717,"style":2313},[65],[27,9719,4672],{"className":9720},[70],[27,9722,9724,9739],{"className":9723},[30],[27,9725,9727],{"className":9726},[34],[36,9728,9729],{"xmlns":38},[40,9730,9731,9737],{},[43,9732,9733,9735],{},[508,9734,3441],{},[46,9736,2300],{"mathvariant":2299},[50,9738,3472],{"encoding":52},[27,9740,9742],{"className":9741,"ariaHidden":57},[56],[27,9743,9745,9748],{"className":9744},[61],[27,9746],{"className":9747,"style":2313},[65],[27,9749,3472],{"className":9750},[70],". Small pointing errors can shift ",[27,9753,9755,9768],{"className":9754},[30],[27,9756,9758],{"className":9757},[34],[36,9759,9760],{"xmlns":38},[40,9761,9762,9766],{},[43,9763,9764],{},[46,9765,48],{},[50,9767,48],{"encoding":52},[27,9769,9771],{"className":9770,"ariaHidden":57},[56],[27,9772,9774,9777],{"className":9773},[61],[27,9775],{"className":9776,"style":66},[65],[27,9778,48],{"className":9779,"style":72},[70,71]," by metres when angles are ill-conditioned.",[88,9782,9784],{"id":9783},"collinear-control","Collinear control",[15,9786,9787,9788,164,9816,164,9844,9872],{},"Nearly collinear ",[27,9789,9791,9804],{"className":9790},[30],[27,9792,9794],{"className":9793},[34],[36,9795,9796],{"xmlns":38},[40,9797,9798,9802],{},[43,9799,9800],{},[46,9801,149],{},[50,9803,149],{"encoding":52},[27,9805,9807],{"className":9806,"ariaHidden":57},[56],[27,9808,9810,9813],{"className":9809},[61],[27,9811],{"className":9812,"style":66},[65],[27,9814,149],{"className":9815},[70,71],[27,9817,9819,9832],{"className":9818},[30],[27,9820,9822],{"className":9821},[34],[36,9823,9824],{"xmlns":38},[40,9825,9826,9830],{},[43,9827,9828],{},[46,9829,179],{},[50,9831,179],{"encoding":52},[27,9833,9835],{"className":9834,"ariaHidden":57},[56],[27,9836,9838,9841],{"className":9837},[61],[27,9839],{"className":9840,"style":66},[65],[27,9842,179],{"className":9843,"style":194},[70,71],[27,9845,9847,9860],{"className":9846},[30],[27,9848,9850],{"className":9849},[34],[36,9851,9852],{"xmlns":38},[40,9853,9854,9858],{},[43,9855,9856],{},[46,9857,210],{},[50,9859,210],{"encoding":52},[27,9861,9863],{"className":9862,"ariaHidden":57},[56],[27,9864,9866,9869],{"className":9865},[61],[27,9867],{"className":9868,"style":66},[65],[27,9870,210],{"className":9871,"style":225},[70,71]," collapses triangle area and makes side-of-line ambiguity severe.",[88,9874,9876],{"id":9875},"no-redundancy","No redundancy",[15,9878,9879],{},"Three angles solve two unknowns exactly — no redundancy to catch misidentified targets or disturbed control. Observe a fourth point for least-squares adjustment when possible.",[10,9881,9883],{"id":9882},"next-steps","Next Steps",[15,9885,9886,9887,485],{},"Tienstra resection establishes 2D control from angles alone — elegant and field-practical when geometry is strong and the danger circle is avoided. For rapid computation without manual cotangent steps, use the ",[75,9888,78],{"href":77},{"title":3490,"searchDepth":9890,"depth":9890,"links":9891},2,[9892,9893,9898,9899,9904,9909,9915,9922,9926,9931],{"id":12,"depth":9890,"text":13},{"id":82,"depth":9890,"text":83,"children":9894},[9895,9897],{"id":90,"depth":9896,"text":91},3,{"id":97,"depth":9896,"text":98},{"id":419,"depth":9890,"text":420},{"id":2047,"depth":9890,"text":2048,"children":9900},[9901,9902,9903],{"id":2054,"depth":9896,"text":2055},{"id":2870,"depth":9896,"text":2871},{"id":3220,"depth":9896,"text":3221},{"id":3493,"depth":9890,"text":3494,"children":9905},[9906,9907,9908],{"id":3497,"depth":9896,"text":3498},{"id":3512,"depth":9896,"text":3513},{"id":4339,"depth":9896,"text":4340},{"id":4346,"depth":9890,"text":4347,"children":9910},[9911,9912,9913],{"id":4350,"depth":9896,"text":4351},{"id":4874,"depth":9896,"text":4875},{"id":5727,"depth":9896,"text":9914},"Step 3: Coordinates of PPP",{"id":6920,"depth":9890,"text":6921,"children":9916},[9917,9918,9920,9921],{"id":6924,"depth":9896,"text":6925},{"id":7501,"depth":9896,"text":9919},"Interior angles of △ABC\\triangle ABC△ABC",{"id":8391,"depth":9896,"text":8392},{"id":9153,"depth":9896,"text":9154},{"id":9349,"depth":9890,"text":9350,"children":9923},[9924,9925],{"id":9353,"depth":9896,"text":9354},{"id":9547,"depth":9896,"text":9548},{"id":9682,"depth":9890,"text":9683,"children":9927},[9928,9929,9930],{"id":9686,"depth":9896,"text":9687},{"id":9783,"depth":9896,"text":9784},{"id":9875,"depth":9896,"text":9876},{"id":9882,"depth":9890,"text":9883},"tools","Three-point angular resection — Tienstra barycentric method, danger circle, field practice, worked example, and verification.","md",{},true,"\u002Flearn\u002Ftools\u002Ftienstra-resection",{"title":5,"description":9933},{"loc":9937},"learn\u002Ftools\u002Ftienstra-resection","KKNmMPhJCVWQct9uWXqsB7Mr5U3vo23rUHK8hTAyCzA",[9943,9948],{"title":9944,"path":9945,"stem":9946,"description":9947,"category":9932,"children":-1},"Traverse Closure & Bowditch Adjustment","\u002Flearn\u002Ftools\u002Ftraverse-and-bowditch","learn\u002Ftools\u002Ftraverse-and-bowditch","Closed traverses — departures and latitudes, linear misclosure, relative precision, Bowditch compass-rule adjustment, and a full worked example.",{"title":9949,"path":9950,"stem":9951,"description":9952,"category":9932,"children":-1},"Polygon Area","\u002Flearn\u002Ftools\u002Fpolygon-area","learn\u002Ftools\u002Fpolygon-area","Planimetric area and perimeter from ordered E\u002FN vertices — the Shoelace formula, winding order, worked example, and common pitfalls.",1783851946152]