3877:
2627:
3063:
2229:
2634:
1555:
3618:
798:
2622:{\displaystyle {\begin{aligned}&s(x)=\sin ^{2}x\\&A(x,y,z)=(x+y+z)^{2}-2(x^{2}+y^{2}+z^{2})\\&{\begin{alignedat}{5}r_{1}&=s(\beta )&\qquad Q_{1}&={\overline {BC}}^{2}\\r_{2}&=s(\alpha )&Q_{2}&={\overline {AC}}^{2}\\r_{3}&=s(\alpha +\beta )&Q_{3}&={\overline {AB}}^{2}\end{alignedat}}\end{aligned}}}
3377:
1718:
1886:
1362:
3386:
571:
3058:{\displaystyle {\begin{aligned}R_{1}&={\frac {r_{2}Q_{3}}{r_{3}}}\qquad R_{2}={\frac {r_{1}Q_{3}}{r_{3}}}\\C_{0}&={\frac {(Q_{1}+Q_{2}+Q_{3})(r_{1}+r_{2}+r_{3})-2(Q_{1}r_{1}+Q_{2}r_{2}+Q_{3}r_{3})}{2r_{3}}}\\D_{0}&={\frac {r_{1}r_{2}A(Q_{1},Q_{2},Q_{3})}{r_{3}}}\end{aligned}}}
391:
931:
1080:
3147:
1562:
1550:{\displaystyle {\overline {PC}}={\begin{cases}{\dfrac {{\overline {BC}}\sin y}{\sin \beta }}&{\text{if }}|\sin \beta |>|\sin \alpha |,\\{\dfrac {{\overline {AC}}\sin x}{\sin \alpha }}&{\text{otherwise.}}\end{cases}}}
1730:
477:
3624:
Ventura et al. solve the planar and three-dimensional
Snellius-Pothenot problem via Vector Geometric Algebra and Conformal Geometric Algebra. The authors also characterize the solutions' sensitivity to measurement errors.
2154:
273:
3613:{\displaystyle {\begin{aligned}{\overline {AP}}^{2}&={\frac {v_{1}R_{1}}{r_{2}}}={\frac {v_{1}Q_{3}}{r_{3}}}\\{\overline {BP}}^{2}&={\frac {v_{2}R_{2}}{r_{1}}}={\frac {v_{2}Q_{3}}{r_{3}}}\end{aligned}}}
807:
793:{\displaystyle \tan \left({\tfrac {\pi }{4}}-\phi \right)={\frac {1-\tan \phi }{\tan \phi +1}}\ ,\qquad {\frac {\tan {\tfrac {1}{2}}(x-y)}{\tan {\tfrac {1}{2}}(x+y)}}={\frac {\sin x-\sin y}{\sin x+\sin y}}\ ,}
2639:
2234:
3391:
3152:
1355:
3762:
558:
481:(A minor note: one should be concerned about division by zero, but consider that the problem is symmetric, so if one of the two given angles is zero one can, if needed, rename that angle
194:
1159:
3674:
is known as the "danger circle", and observations made on (or very close to) this circle should be avoided. It is helpful to plot this circle on a map before making the observations.
3140:
1279:
976:
2210:
The following solution is based upon a paper by N. J. Wildberger. It has the advantage that it is almost purely algebraic. The only place trigonometry is used is in converting the
2383:
1981:
1935:
404:
3372:{\displaystyle {\begin{aligned}v_{1}&=1-{\frac {(R_{1}+R_{3}-Q_{2})^{2}}{4R_{1}R_{3}}}\\v_{2}&=1-{\frac {(R_{2}+R_{3}-Q_{1})^{2}}{4R_{2}R_{3}}}\end{aligned}}}
497:
as well. This will suffice to guarantee that the ratio above is well defined. An alternative approach to the zero angle problem is given in the algorithm below.)
1713:{\displaystyle {\overline {PA}}={\sqrt {{\overline {AC}}^{2}+{\overline {PC}}^{2}-2\cdot {\overline {AC}}\cdot {\overline {PC}}\cdot \cos(\pi -\alpha -x)}}.}
1881:{\displaystyle {\overline {PB}}={\sqrt {{\overline {BC}}^{2}+{\overline {PC}}^{2}-2\cdot {\overline {BC}}\cdot {\overline {PC}}\cdot \cos(\pi -\beta -y)}}.}
3972:
Ventura, Jorge; Martinez, Fernando; Manzano-Agugliaro, Francisco; Návrat, Aleš; Hrdina, Jaroslav; Eid, Ahmad H.; Montoya, Francisco G. (2024-05-27).
3947:
2088:
3764:. If this condition is observed the computer/spreadsheet calculations should be stopped and an error message ("indeterminate case") returned.
505:
99:
20:
3903:(1650–1732) did not deserve to be included as he had made no original contribution, but merely restated Snellius 75 years later.
146:
1286:
386:{\displaystyle {\overline {PC}}={\frac {{\overline {AC}}\sin x}{\sin \alpha }}={\frac {{\overline {BC}}\sin y}{\sin \beta }}.}
4105:
3079:
3974:"A novel geometric method based on conformal geometric algebra applied to the resection problem in two and three dimensions"
3696:
926:{\displaystyle \tan {\tfrac {1}{2}}(x-y)=\tan {\tfrac {1}{2}}(\alpha +\beta +C)\tan \left({\tfrac {\pi }{4}}-\phi \right).}
4063:, Journal of Applied Geodesy, Band 13, Heft 1, Seiten 27–31, ISSN (Online) 1862-9024, ISSN (Print) 1862-9016, DOI:
3917:
1114:
1075:{\displaystyle \phi ={\mathsf {atan2}}\left({\overline {BC}}\sin \alpha ,\ {\overline {AC}}\sin \beta \right),}
1166:
2000:
2186:
at the intersection of the two circles (the two circles intersect at two points; one intersection point is
1389:
3885:
563:
4095:
3912:
2215:
3900:
3876:
98:
Since it involves the observation of known points from an unknown point, the problem is an example of
3678:
3861:
A slightly more challenging test case for a computer program uses the same data but this time with
1940:
1894:
103:
4100:
4043:
3995:
1984:
943:
are known the various triangles can be solved straightforwardly to determine the position of
4035:
3985:
3641:, the problem has an infinite number of solutions; the reason is that from any other point
3664:
1097:
55:
3382:
1721:
3948:"Greek Geometry, Rational Trigonometry, and the Snellius – Pothenot Surveying Problem"
4089:
199:
4061:
Analysing
Willerding’s formula for solving the planar three point resection problem
472:{\displaystyle \tan \phi ={\frac {{\rm {BC}}\sin \alpha }{{\rm {AC}}\sin \beta }}.}
245:
47:
27:
4080:
2219:
3990:
3868: = 0. The program should return the answers 843, 1157 and 837.
1093:
19:
3999:
3973:
35:
1108:
function correctly handles the case where one of the two arguments is zero.
2149:{\displaystyle {\overline {MO}}={\tfrac {\overline {AC}}{2\tan \alpha }}.}
1089:
4064:
1092:
function, also called the arctangent of two arguments, that returns the
4047:
3681:
is helpful in detecting the indeterminate situation. The quadrilateral
209:
represents the (known) internal angle in this quadrilateral at point
4039:
4023:
3689:
a pair of opposite angles (such as the angle at P and the angle at
3875:
2211:
1084:
59:
3783:
3837:, a different configuration from the one shown in the figure).
3686:
3667:). Thus the solution in this case is not uniquely determined.
1100:
the two arguments are reversed, so the proper syntax would be
935:
Now these two equations in two unknowns must be solved. Once
395:
A useful trick at this point is to define an auxiliary angle
18:
1543:
2053:
Therefore, on a map or nautical chart showing the points
1350:{\displaystyle x={\frac {K+W}{2}},\ y={\frac {K-W}{2}}.}
75:; the problem is to determine the position of the point
3888:(1870–1942) wrote that the proper term in English was
3757:{\displaystyle \alpha +\beta +C=k\pi ,(k=1,2,\cdots )}
2111:
1231:
1203:
898:
854:
818:
702:
667:
587:
34:
is a problem first described in the context of planar
3699:
3389:
3150:
3082:
3067:
the following equation gives two possible values for
2637:
2232:
2091:
1943:
1897:
1733:
1565:
1491:
1393:
1365:
1289:
1169:
1117:
979:
810:
574:
508:
407:
276:
149:
2057:, the following graphical construction can be used:
553:{\displaystyle {\frac {\sin x}{\sin y}}=\tan \phi .}
4079:, Washington D.C., Heath & Co., 1892, page 188
1096:of the ratio of the two values given. Note that in
3756:
3612:
3371:
3134:
3057:
2621:
2148:
1975:
1929:
1880:
1712:
1549:
1349:
1273:
1153:
1074:
925:
792:
552:
471:
385:
188:
3651:of this circle the observer sees the same angles
2013:is a circle having its center on the midline of
802:to put this in the form of the second equation;
3817: = 15 degrees. Find the distances of
3772:(Adapted form Bowser, exercise 140, page 203).
198:by using the sum of the angles formula for the
4077:A treatise on plane and spherical trigonometry
8:
3637:happens to be located on the same circle as
2198:This method of solution is sometimes called
500:With this substitution the equation becomes
3799: = 255.8 degrees. From a station
2036:. Similarly the locus of points from which
2050:is at the intersection of these two loci.
213:. (Note that in the case where the points
189:{\displaystyle x+y=2\pi -\alpha -\beta -C}
3989:
3698:
3598:
3587:
3577:
3570:
3559:
3548:
3538:
3531:
3518:
3503:
3490:
3479:
3469:
3462:
3451:
3440:
3430:
3423:
3410:
3395:
3390:
3388:
3356:
3346:
3331:
3321:
3308:
3295:
3285:
3266:
3249:
3239:
3224:
3214:
3201:
3188:
3178:
3159:
3151:
3149:
3144:choosing the larger of these values, let:
3126:
3113:
3103:
3090:
3081:
3043:
3029:
3016:
3003:
2987:
2977:
2970:
2957:
2940:
2922:
2912:
2899:
2889:
2876:
2866:
2844:
2831:
2818:
2802:
2789:
2776:
2766:
2753:
2737:
2726:
2716:
2709:
2700:
2687:
2676:
2666:
2659:
2646:
2638:
2636:
2605:
2590:
2576:
2539:
2525:
2510:
2496:
2465:
2451:
2436:
2422:
2390:
2382:
2368:
2355:
2342:
2323:
2257:
2233:
2231:
2171:Repeat the same construction with points
2110:
2092:
2090:
1967:
1954:
1942:
1921:
1908:
1896:
1828:
1810:
1795:
1780:
1770:
1755:
1752:
1734:
1732:
1660:
1642:
1627:
1612:
1602:
1587:
1584:
1566:
1564:
1535:
1494:
1490:
1478:
1464:
1456:
1442:
1437:
1396:
1392:
1384:
1366:
1364:
1326:
1296:
1288:
1230:
1223:
1202:
1168:
1154:{\displaystyle K=2\pi -\alpha -\beta -C.}
1116:
1040:
1010:
987:
986:
978:
947:. The detailed procedure is shown below.
897:
853:
817:
809:
734:
701:
666:
657:
612:
586:
573:
509:
507:
445:
444:
424:
423:
420:
406:
342:
339:
298:
295:
277:
275:
148:
1983:are known in some appropriate Cartesian
3929:
3135:{\displaystyle (R_{3}-C_{0})^{2}=D_{0}}
1274:{\displaystyle W=2\cdot \arctan \left.}
102:. Historically it was first studied by
1102:= atan2(AC*\sin(beta), BC*\sin(alpha))
1000:
997:
994:
991:
988:
3825:. (Note that in this case the points
2046:is another circle. The desired point
7:
3896:was the continental European usage.
969:, the solution proceeds as follows.
485:and call the other (non-zero) angle
106:, who found a solution around 1615.
3880:Plaque on Snellius' house in Leiden
2190:and the other is the desired point
3884:The British authority on geodesy,
449:
446:
428:
425:
42:, an observer at an unknown point
14:
3833:are on the same side of the line
221:are on the same side of the line
79:. (See figure; the point denoted
3955:Chamchuri Journal of Mathematics
2695:
2417:
2156:Draw the circle with center at
2071:and the midline, which crosses
2003:the locus of points from which
656:
3751:
3727:
3620:Solution via Geometric Algebra
3328:
3288:
3221:
3181:
3110:
3083:
3035:
2996:
2928:
2859:
2850:
2811:
2808:
2769:
2567:
2555:
2487:
2481:
2412:
2406:
2374:
2335:
2320:
2301:
2295:
2277:
2247:
2241:
2206:Rational trigonometry approach
2081:. On this line find the point
1995:Geometric (graphical) solution
1870:
1852:
1702:
1684:
1479:
1465:
1457:
1443:
1260:
1242:
1220:
1199:
883:
865:
841:
829:
725:
713:
690:
678:
119:Denoting the (unknown) angles
1:
3946:Norman J. Wildberger (2010).
3693:) are supplementary i.e. iff
1976:{\displaystyle C:x_{C},y_{C}}
1930:{\displaystyle A:x_{A},y_{A}}
3810: = 30 degrees and
3776:are three objects such that
3513:
3405:
2600:
2520:
2446:
2122:
2102:
1838:
1820:
1790:
1765:
1744:
1670:
1652:
1622:
1597:
1576:
1504:
1406:
1376:
1050:
1020:
352:
308:
287:
38:. Given three known points
4122:
3991:10.1007/s00190-024-01854-1
3899:McCaw thought the name of
3918:Triangulation (surveying)
489:, reversing the roles of
110:Formulating the equations
32:Snellius–Pothenot problem
4028:The Geographical Journal
1987:then the coordinates of
564:trigonometric identities
3665:inscribed angle theorem
2001:inscribed angle theorem
268:in two different ways:
16:Problem in trigonometry
3881:
3792: = 320, and
3758:
3629:The indeterminate case
3614:
3373:
3136:
3059:
2623:
2150:
1991:can be found as well.
1977:
1931:
1891:If the coordinates of
1882:
1714:
1551:
1351:
1275:
1155:
1076:
955:Given are two lengths
927:
794:
554:
473:
387:
190:
23:
4106:Mathematical problems
4024:"Resection in Survey"
4022:McCaw, G. T. (1918).
3913:Solution of triangles
3879:
3759:
3679:cyclic quadrilaterals
3615:
3374:
3137:
3060:
2624:
2226:define the following:
2151:
1978:
1932:
1883:
1720:(This comes from the
1715:
1552:
1352:
1276:
1156:
1077:
928:
795:
555:
474:
388:
232:will be greater than
191:
22:
3886:George Tyrrell McCaw
3803:it is observed that
3697:
3387:
3148:
3080:
2635:
2230:
2218:. There is only one
2089:
1941:
1895:
1731:
1563:
1363:
1287:
1167:
1115:
977:
808:
572:
566:can be used, namely
506:
405:
274:
147:
3670:The circle through
3645:located on the arc
2077:perpendicularly at
965:, and three angles
4075:Edward A. Bowser:
4012:Bowser: A treatise
3978:Journal of Geodesy
3936:Bowser: A treatise
3882:
3872:Naming controversy
3858: = 502.
3852: = 777,
3846: = 790,
3782: = 435 (
3754:
3610:
3608:
3369:
3367:
3132:
3055:
3053:
2619:
2617:
2613:
2146:
2141:
2042:subtends an angle
2029:subtends an angle
2019:; from the center
2009:subtends an angle
1973:
1927:
1878:
1710:
1547:
1542:
1531:
1433:
1347:
1271:
1240:
1212:
1151:
1072:
951:Solution algorithm
923:
907:
863:
827:
790:
711:
676:
596:
550:
469:
383:
186:
71:subtends an angle
46:observes that the
24:
3894:Snellius-Pothenot
3604:
3565:
3516:
3496:
3457:
3408:
3363:
3256:
3049:
2947:
2743:
2693:
2603:
2523:
2449:
2140:
2125:
2105:
2061:Draw the segment
1985:coordinate system
1873:
1841:
1823:
1793:
1768:
1747:
1705:
1673:
1655:
1625:
1600:
1579:
1538:
1530:
1507:
1440:
1432:
1409:
1379:
1342:
1319:
1312:
1239:
1211:
1053:
1039:
1023:
967:α, β, C
906:
862:
826:
786:
782:
729:
710:
675:
652:
648:
595:
533:
464:
378:
355:
334:
311:
290:
262:, we can express
4113:
4059:Gerhard Heindl:
4052:
4051:
4019:
4013:
4010:
4004:
4003:
3993:
3969:
3963:
3962:
3952:
3943:
3937:
3934:
3901:Laurent Pothenot
3890:Snellius problem
3867:
3857:
3856:
3851:
3850:
3845:
3844:
3836:
3832:
3828:
3824:
3820:
3816:
3809:
3802:
3798:
3791:
3790:
3781:
3780:
3775:
3763:
3761:
3760:
3755:
3692:
3684:
3673:
3662:
3658:
3654:
3650:
3649:
3644:
3640:
3636:
3619:
3617:
3616:
3611:
3609:
3605:
3603:
3602:
3593:
3592:
3591:
3582:
3581:
3571:
3566:
3564:
3563:
3554:
3553:
3552:
3543:
3542:
3532:
3523:
3522:
3517:
3512:
3504:
3497:
3495:
3494:
3485:
3484:
3483:
3474:
3473:
3463:
3458:
3456:
3455:
3446:
3445:
3444:
3435:
3434:
3424:
3415:
3414:
3409:
3404:
3396:
3378:
3376:
3375:
3370:
3368:
3364:
3362:
3361:
3360:
3351:
3350:
3337:
3336:
3335:
3326:
3325:
3313:
3312:
3300:
3299:
3286:
3271:
3270:
3257:
3255:
3254:
3253:
3244:
3243:
3230:
3229:
3228:
3219:
3218:
3206:
3205:
3193:
3192:
3179:
3164:
3163:
3141:
3139:
3138:
3133:
3131:
3130:
3118:
3117:
3108:
3107:
3095:
3094:
3075:
3064:
3062:
3061:
3056:
3054:
3050:
3048:
3047:
3038:
3034:
3033:
3021:
3020:
3008:
3007:
2992:
2991:
2982:
2981:
2971:
2962:
2961:
2948:
2946:
2945:
2944:
2931:
2927:
2926:
2917:
2916:
2904:
2903:
2894:
2893:
2881:
2880:
2871:
2870:
2849:
2848:
2836:
2835:
2823:
2822:
2807:
2806:
2794:
2793:
2781:
2780:
2767:
2758:
2757:
2744:
2742:
2741:
2732:
2731:
2730:
2721:
2720:
2710:
2705:
2704:
2694:
2692:
2691:
2682:
2681:
2680:
2671:
2670:
2660:
2651:
2650:
2628:
2626:
2625:
2620:
2618:
2614:
2610:
2609:
2604:
2599:
2591:
2581:
2580:
2544:
2543:
2530:
2529:
2524:
2519:
2511:
2501:
2500:
2470:
2469:
2456:
2455:
2450:
2445:
2437:
2427:
2426:
2395:
2394:
2380:
2373:
2372:
2360:
2359:
2347:
2346:
2328:
2327:
2272:
2262:
2261:
2236:
2200:Cassini's method
2193:
2189:
2185:
2178:
2174:
2167:
2163:
2160:passing through
2159:
2155:
2153:
2152:
2147:
2142:
2139:
2121:
2113:
2112:
2106:
2101:
2093:
2084:
2080:
2076:
2075:
2070:
2066:
2065:
2056:
2049:
2045:
2041:
2040:
2035:
2028:
2027:
2023:of this circle,
2022:
2018:
2017:
2012:
2008:
2007:
1990:
1982:
1980:
1979:
1974:
1972:
1971:
1959:
1958:
1936:
1934:
1933:
1928:
1926:
1925:
1913:
1912:
1887:
1885:
1884:
1879:
1874:
1842:
1837:
1829:
1824:
1819:
1811:
1800:
1799:
1794:
1789:
1781:
1775:
1774:
1769:
1764:
1756:
1753:
1748:
1743:
1735:
1719:
1717:
1716:
1711:
1706:
1674:
1669:
1661:
1656:
1651:
1643:
1632:
1631:
1626:
1621:
1613:
1607:
1606:
1601:
1596:
1588:
1585:
1580:
1575:
1567:
1556:
1554:
1553:
1548:
1546:
1545:
1539:
1536:
1532:
1529:
1518:
1508:
1503:
1495:
1492:
1482:
1468:
1460:
1446:
1441:
1438:
1434:
1431:
1420:
1410:
1405:
1397:
1394:
1380:
1375:
1367:
1356:
1354:
1353:
1348:
1343:
1338:
1327:
1317:
1313:
1308:
1297:
1280:
1278:
1277:
1272:
1267:
1263:
1241:
1232:
1213:
1204:
1160:
1158:
1157:
1152:
1107:
1103:
1087:
1081:
1079:
1078:
1073:
1068:
1064:
1054:
1049:
1041:
1037:
1024:
1019:
1011:
1004:
1003:
968:
964:
963:
959:
946:
942:
938:
932:
930:
929:
924:
919:
915:
908:
899:
864:
855:
828:
819:
799:
797:
796:
791:
784:
783:
781:
758:
735:
730:
728:
712:
703:
693:
677:
668:
658:
650:
649:
647:
630:
613:
608:
604:
597:
588:
559:
557:
556:
551:
534:
532:
521:
510:
496:
492:
488:
484:
478:
476:
475:
470:
465:
463:
453:
452:
442:
432:
431:
421:
398:
392:
390:
389:
384:
379:
377:
366:
356:
351:
343:
340:
335:
333:
322:
312:
307:
299:
296:
291:
286:
278:
267:
266:
261:
254:
235:
231:
224:
220:
216:
212:
208:
204:
195:
193:
192:
187:
140:
136:
129:
125:
94:
90:
86:
82:
78:
74:
70:
69:
65:and the segment
64:
54:
53:
45:
41:
4121:
4120:
4116:
4115:
4114:
4112:
4111:
4110:
4086:
4085:
4072:
4056:
4055:
4040:10.2307/1779558
4021:
4020:
4016:
4011:
4007:
3971:
3970:
3966:
3950:
3945:
3944:
3940:
3935:
3931:
3926:
3909:
3874:
3862:
3854:
3853:
3848:
3847:
3842:
3841:
3834:
3830:
3826:
3822:
3818:
3811:
3804:
3800:
3793:
3788:
3787:
3778:
3777:
3773:
3770:
3768:Solved examples
3695:
3694:
3690:
3682:
3671:
3660:
3656:
3652:
3647:
3646:
3642:
3638:
3634:
3633:When the point
3631:
3622:
3607:
3606:
3594:
3583:
3573:
3572:
3555:
3544:
3534:
3533:
3524:
3505:
3502:
3499:
3498:
3486:
3475:
3465:
3464:
3447:
3436:
3426:
3425:
3416:
3397:
3394:
3385:
3384:
3366:
3365:
3352:
3342:
3338:
3327:
3317:
3304:
3291:
3287:
3272:
3262:
3259:
3258:
3245:
3235:
3231:
3220:
3210:
3197:
3184:
3180:
3165:
3155:
3146:
3145:
3122:
3109:
3099:
3086:
3078:
3077:
3074:
3068:
3052:
3051:
3039:
3025:
3012:
2999:
2983:
2973:
2972:
2963:
2953:
2950:
2949:
2936:
2932:
2918:
2908:
2895:
2885:
2872:
2862:
2840:
2827:
2814:
2798:
2785:
2772:
2768:
2759:
2749:
2746:
2745:
2733:
2722:
2712:
2711:
2696:
2683:
2672:
2662:
2661:
2652:
2642:
2633:
2632:
2616:
2615:
2612:
2611:
2592:
2589:
2582:
2572:
2570:
2545:
2535:
2532:
2531:
2512:
2509:
2502:
2492:
2490:
2471:
2461:
2458:
2457:
2438:
2435:
2428:
2418:
2415:
2396:
2386:
2378:
2377:
2364:
2351:
2338:
2319:
2270:
2269:
2253:
2228:
2227:
2208:
2191:
2187:
2183:
2176:
2172:
2165:
2161:
2157:
2126:
2114:
2094:
2087:
2086:
2082:
2078:
2073:
2072:
2068:
2067:, the midpoint
2063:
2062:
2054:
2047:
2043:
2038:
2037:
2030:
2025:
2024:
2020:
2015:
2014:
2010:
2005:
2004:
1997:
1988:
1963:
1950:
1939:
1938:
1917:
1904:
1893:
1892:
1830:
1812:
1782:
1779:
1757:
1754:
1736:
1729:
1728:
1662:
1644:
1614:
1611:
1589:
1586:
1568:
1561:
1560:
1541:
1540:
1533:
1519:
1496:
1493:
1487:
1486:
1435:
1421:
1398:
1395:
1385:
1368:
1361:
1360:
1328:
1298:
1285:
1284:
1192:
1188:
1165:
1164:
1113:
1112:
1105:
1101:
1098:Microsoft Excel
1083:
1042:
1012:
1009:
1005:
975:
974:
966:
961:
957:
956:
953:
944:
940:
936:
896:
892:
806:
805:
759:
736:
694:
659:
631:
614:
585:
581:
570:
569:
522:
511:
504:
503:
494:
490:
486:
482:
443:
422:
403:
402:
396:
367:
344:
341:
323:
300:
297:
279:
272:
271:
264:
263:
256:
249:
242:
240:Second equation
233:
226:
222:
218:
214:
210:
206:
205:. The variable
202:
145:
144:
138:
131:
127:
120:
117:
112:
92:
88:
84:
80:
76:
72:
67:
66:
62:
51:
50:
43:
39:
17:
12:
11:
5:
4119:
4117:
4109:
4108:
4103:
4098:
4088:
4087:
4084:
4083:
4071:
4068:
4067:
4066:
4054:
4053:
4034:(2): 105–126.
4014:
4005:
3964:
3938:
3928:
3927:
3925:
3922:
3921:
3920:
3915:
3908:
3905:
3873:
3870:
3769:
3766:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3630:
3627:
3621:
3601:
3597:
3590:
3586:
3580:
3576:
3569:
3562:
3558:
3551:
3547:
3541:
3537:
3530:
3527:
3525:
3521:
3515:
3511:
3508:
3501:
3500:
3493:
3489:
3482:
3478:
3472:
3468:
3461:
3454:
3450:
3443:
3439:
3433:
3429:
3422:
3419:
3417:
3413:
3407:
3403:
3400:
3393:
3392:
3381:
3380:
3379:
3359:
3355:
3349:
3345:
3341:
3334:
3330:
3324:
3320:
3316:
3311:
3307:
3303:
3298:
3294:
3290:
3284:
3281:
3278:
3275:
3273:
3269:
3265:
3261:
3260:
3252:
3248:
3242:
3238:
3234:
3227:
3223:
3217:
3213:
3209:
3204:
3200:
3196:
3191:
3187:
3183:
3177:
3174:
3171:
3168:
3166:
3162:
3158:
3154:
3153:
3142:
3129:
3125:
3121:
3116:
3112:
3106:
3102:
3098:
3093:
3089:
3085:
3072:
3065:
3046:
3042:
3037:
3032:
3028:
3024:
3019:
3015:
3011:
3006:
3002:
2998:
2995:
2990:
2986:
2980:
2976:
2969:
2966:
2964:
2960:
2956:
2952:
2951:
2943:
2939:
2935:
2930:
2925:
2921:
2915:
2911:
2907:
2902:
2898:
2892:
2888:
2884:
2879:
2875:
2869:
2865:
2861:
2858:
2855:
2852:
2847:
2843:
2839:
2834:
2830:
2826:
2821:
2817:
2813:
2810:
2805:
2801:
2797:
2792:
2788:
2784:
2779:
2775:
2771:
2765:
2762:
2760:
2756:
2752:
2748:
2747:
2740:
2736:
2729:
2725:
2719:
2715:
2708:
2703:
2699:
2690:
2686:
2679:
2675:
2669:
2665:
2658:
2655:
2653:
2649:
2645:
2641:
2640:
2629:
2608:
2602:
2598:
2595:
2588:
2585:
2583:
2579:
2575:
2571:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2546:
2542:
2538:
2534:
2533:
2528:
2522:
2518:
2515:
2508:
2505:
2503:
2499:
2495:
2491:
2489:
2486:
2483:
2480:
2477:
2474:
2472:
2468:
2464:
2460:
2459:
2454:
2448:
2444:
2441:
2434:
2431:
2429:
2425:
2421:
2416:
2414:
2411:
2408:
2405:
2402:
2399:
2397:
2393:
2389:
2385:
2384:
2381:
2379:
2376:
2371:
2367:
2363:
2358:
2354:
2350:
2345:
2341:
2337:
2334:
2331:
2326:
2322:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2271:
2268:
2265:
2260:
2256:
2252:
2249:
2246:
2243:
2240:
2237:
2235:
2207:
2204:
2196:
2195:
2180:
2175:and the angle
2169:
2145:
2138:
2135:
2132:
2129:
2124:
2120:
2117:
2109:
2104:
2100:
2097:
1996:
1993:
1970:
1966:
1962:
1957:
1953:
1949:
1946:
1924:
1920:
1916:
1911:
1907:
1903:
1900:
1889:
1888:
1877:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1840:
1836:
1833:
1827:
1822:
1818:
1815:
1809:
1806:
1803:
1798:
1792:
1788:
1785:
1778:
1773:
1767:
1763:
1760:
1751:
1746:
1742:
1739:
1725:
1722:law of cosines
1709:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1672:
1668:
1665:
1659:
1654:
1650:
1647:
1641:
1638:
1635:
1630:
1624:
1620:
1617:
1610:
1605:
1599:
1595:
1592:
1583:
1578:
1574:
1571:
1557:
1544:
1534:
1528:
1525:
1522:
1517:
1514:
1511:
1506:
1502:
1499:
1489:
1488:
1485:
1481:
1477:
1474:
1471:
1467:
1463:
1459:
1455:
1452:
1449:
1445:
1436:
1430:
1427:
1424:
1419:
1416:
1413:
1408:
1404:
1401:
1391:
1390:
1388:
1383:
1378:
1374:
1371:
1357:
1346:
1341:
1337:
1334:
1331:
1325:
1322:
1316:
1311:
1307:
1304:
1301:
1295:
1292:
1281:
1270:
1266:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1238:
1235:
1229:
1226:
1222:
1219:
1216:
1210:
1207:
1201:
1198:
1195:
1191:
1187:
1184:
1181:
1178:
1175:
1172:
1161:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1109:
1071:
1067:
1063:
1060:
1057:
1052:
1048:
1045:
1036:
1033:
1030:
1027:
1022:
1018:
1015:
1008:
1002:
999:
996:
993:
990:
985:
982:
952:
949:
922:
918:
914:
911:
905:
902:
895:
891:
888:
885:
882:
879:
876:
873:
870:
867:
861:
858:
852:
849:
846:
843:
840:
837:
834:
831:
825:
822:
816:
813:
789:
780:
777:
774:
771:
768:
765:
762:
757:
754:
751:
748:
745:
742:
739:
733:
727:
724:
721:
718:
715:
709:
706:
700:
697:
692:
689:
686:
683:
680:
674:
671:
665:
662:
655:
646:
643:
640:
637:
634:
629:
626:
623:
620:
617:
611:
607:
603:
600:
594:
591:
584:
580:
577:
562:Now two known
549:
546:
543:
540:
537:
531:
528:
525:
520:
517:
514:
468:
462:
459:
456:
451:
448:
441:
438:
435:
430:
427:
419:
416:
413:
410:
382:
376:
373:
370:
365:
362:
359:
354:
350:
347:
338:
332:
329:
326:
321:
318:
315:
310:
306:
303:
294:
289:
285:
282:
241:
238:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
116:
115:First equation
113:
111:
108:
15:
13:
10:
9:
6:
4:
3:
2:
4118:
4107:
4104:
4102:
4099:
4097:
4094:
4093:
4091:
4082:
4078:
4074:
4073:
4069:
4065:
4062:
4058:
4057:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4018:
4015:
4009:
4006:
4001:
3997:
3992:
3987:
3983:
3979:
3975:
3968:
3965:
3960:
3956:
3949:
3942:
3939:
3933:
3930:
3923:
3919:
3916:
3914:
3911:
3910:
3906:
3904:
3902:
3897:
3895:
3891:
3887:
3878:
3871:
3869:
3866:
3859:
3838:
3815:
3808:
3797:
3785:
3767:
3765:
3748:
3745:
3742:
3739:
3736:
3733:
3730:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3688:
3680:
3677:A theorem on
3675:
3668:
3666:
3628:
3626:
3599:
3595:
3588:
3584:
3578:
3574:
3567:
3560:
3556:
3549:
3545:
3539:
3535:
3528:
3526:
3519:
3509:
3506:
3491:
3487:
3480:
3476:
3470:
3466:
3459:
3452:
3448:
3441:
3437:
3431:
3427:
3420:
3418:
3411:
3401:
3398:
3357:
3353:
3347:
3343:
3339:
3332:
3322:
3318:
3314:
3309:
3305:
3301:
3296:
3292:
3282:
3279:
3276:
3274:
3267:
3263:
3250:
3246:
3240:
3236:
3232:
3225:
3215:
3211:
3207:
3202:
3198:
3194:
3189:
3185:
3175:
3172:
3169:
3167:
3160:
3156:
3143:
3127:
3123:
3119:
3114:
3104:
3100:
3096:
3091:
3087:
3071:
3066:
3044:
3040:
3030:
3026:
3022:
3017:
3013:
3009:
3004:
3000:
2993:
2988:
2984:
2978:
2974:
2967:
2965:
2958:
2954:
2941:
2937:
2933:
2923:
2919:
2913:
2909:
2905:
2900:
2896:
2890:
2886:
2882:
2877:
2873:
2867:
2863:
2856:
2853:
2845:
2841:
2837:
2832:
2828:
2824:
2819:
2815:
2803:
2799:
2795:
2790:
2786:
2782:
2777:
2773:
2763:
2761:
2754:
2750:
2738:
2734:
2727:
2723:
2717:
2713:
2706:
2701:
2697:
2688:
2684:
2677:
2673:
2667:
2663:
2656:
2654:
2647:
2643:
2630:
2606:
2596:
2593:
2586:
2584:
2577:
2573:
2564:
2561:
2558:
2552:
2549:
2547:
2540:
2536:
2526:
2516:
2513:
2506:
2504:
2497:
2493:
2484:
2478:
2475:
2473:
2466:
2462:
2452:
2442:
2439:
2432:
2430:
2423:
2419:
2409:
2403:
2400:
2398:
2391:
2387:
2369:
2365:
2361:
2356:
2352:
2348:
2343:
2339:
2332:
2329:
2324:
2316:
2313:
2310:
2307:
2304:
2298:
2292:
2289:
2286:
2283:
2280:
2274:
2266:
2263:
2258:
2254:
2250:
2244:
2238:
2225:
2224:
2223:
2221:
2217:
2213:
2205:
2203:
2201:
2181:
2170:
2143:
2136:
2133:
2130:
2127:
2118:
2115:
2107:
2098:
2095:
2060:
2059:
2058:
2051:
2034:
2002:
1994:
1992:
1986:
1968:
1964:
1960:
1955:
1951:
1947:
1944:
1922:
1918:
1914:
1909:
1905:
1901:
1898:
1875:
1867:
1864:
1861:
1858:
1855:
1849:
1846:
1843:
1834:
1831:
1825:
1816:
1813:
1807:
1804:
1801:
1796:
1786:
1783:
1776:
1771:
1761:
1758:
1749:
1740:
1737:
1726:
1723:
1707:
1699:
1696:
1693:
1690:
1687:
1681:
1678:
1675:
1666:
1663:
1657:
1648:
1645:
1639:
1636:
1633:
1628:
1618:
1615:
1608:
1603:
1593:
1590:
1581:
1572:
1569:
1558:
1526:
1523:
1520:
1515:
1512:
1509:
1500:
1497:
1483:
1475:
1472:
1469:
1461:
1453:
1450:
1447:
1428:
1425:
1422:
1417:
1414:
1411:
1402:
1399:
1386:
1381:
1372:
1369:
1358:
1344:
1339:
1335:
1332:
1329:
1323:
1320:
1314:
1309:
1305:
1302:
1299:
1293:
1290:
1282:
1268:
1264:
1257:
1254:
1251:
1248:
1245:
1236:
1233:
1227:
1224:
1217:
1214:
1208:
1205:
1196:
1193:
1189:
1185:
1182:
1179:
1176:
1173:
1170:
1162:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1110:
1099:
1095:
1091:
1086:
1069:
1065:
1061:
1058:
1055:
1046:
1043:
1034:
1031:
1028:
1025:
1016:
1013:
1006:
983:
980:
972:
971:
970:
950:
948:
933:
920:
916:
912:
909:
903:
900:
893:
889:
886:
880:
877:
874:
871:
868:
859:
856:
850:
847:
844:
838:
835:
832:
823:
820:
814:
811:
803:
800:
787:
778:
775:
772:
769:
766:
763:
760:
755:
752:
749:
746:
743:
740:
737:
731:
722:
719:
716:
707:
704:
698:
695:
687:
684:
681:
672:
669:
663:
660:
653:
644:
641:
638:
635:
632:
627:
624:
621:
618:
615:
609:
605:
601:
598:
592:
589:
582:
578:
575:
567:
565:
560:
547:
544:
541:
538:
535:
529:
526:
523:
518:
515:
512:
501:
498:
479:
466:
460:
457:
454:
439:
436:
433:
417:
414:
411:
408:
400:
393:
380:
374:
371:
368:
363:
360:
357:
348:
345:
336:
330:
327:
324:
319:
316:
313:
304:
301:
292:
283:
280:
269:
260:
253:
248:in triangles
247:
244:Applying the
239:
237:
230:
201:
200:quadrilateral
196:
183:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
150:
142:
135:
124:
114:
109:
107:
105:
101:
96:
91:as seen from
61:
57:
49:
37:
33:
29:
21:
4096:Trigonometry
4081:Google books
4076:
4060:
4031:
4027:
4017:
4008:
3981:
3977:
3967:
3958:
3954:
3941:
3932:
3898:
3893:
3889:
3883:
3864:
3860:
3839:
3813:
3806:
3795:
3771:
3676:
3669:
3632:
3623:
3069:
2209:
2199:
2197:
2052:
2032:
1998:
1890:
954:
934:
804:
801:
568:
561:
502:
499:
480:
401:
394:
270:
258:
251:
246:law of sines
243:
228:
225:, the angle
197:
143:
133:
122:
118:
97:
48:line segment
31:
28:trigonometry
25:
2220:square root
83:is between
4090:Categories
4070:References
3961:(2): 1–14.
3685:is cyclic
2222:required.
2085:such that
1537:otherwise.
1163:calculate
1111:calculate
1094:arctangent
973:calculate
399:such that
4101:Surveying
4000:1432-1394
3984:(6): 47.
3749:⋯
3722:π
3707:β
3701:α
3514:¯
3406:¯
3315:−
3283:−
3208:−
3176:−
3097:−
2854:−
2601:¯
2565:β
2559:α
2521:¯
2485:α
2447:¯
2410:β
2330:−
2264:
2137:α
2134:
2123:¯
2103:¯
1865:−
1862:β
1859:−
1856:π
1850:
1844:⋅
1839:¯
1826:⋅
1821:¯
1808:⋅
1802:−
1791:¯
1766:¯
1745:¯
1697:−
1694:α
1691:−
1688:π
1682:
1676:⋅
1671:¯
1658:⋅
1653:¯
1640:⋅
1634:−
1623:¯
1598:¯
1577:¯
1527:α
1524:
1513:
1505:¯
1476:α
1473:
1454:β
1451:
1429:β
1426:
1415:
1407:¯
1377:¯
1333:−
1252:β
1246:α
1228:
1218:ϕ
1215:−
1206:π
1197:
1186:
1180:⋅
1143:−
1140:β
1137:−
1134:α
1131:−
1128:π
1062:β
1059:
1051:¯
1032:α
1029:
1021:¯
981:ϕ
913:ϕ
910:−
901:π
890:
875:β
869:α
851:
836:−
815:
776:
764:
753:
747:−
741:
699:
685:−
664:
639:ϕ
636:
628:ϕ
625:
619:−
602:ϕ
599:−
590:π
579:
545:ϕ
542:
527:
516:
461:β
458:
440:α
437:
415:ϕ
412:
375:β
372:
361:
353:¯
331:α
328:
317:
309:¯
288:¯
181:−
178:β
175:−
172:α
169:−
166:π
100:resection
36:surveying
3907:See also
3892:, while
3840:Answer:
3659:as from
3383:finally:
2631:now let:
1439:if
1090:computer
104:Snellius
56:subtends
4048:1779558
3823:A, B, C
3774:A, B, C
3639:A, B, C
2216:spreads
2055:A, B, C
1999:By the
141:gives:
40:A, B, C
4046:
3998:
3657:β
3653:α
2212:angles
2177:β
2044:β
2033:α
2011:α
1318:
1183:arctan
1104:. The
1082:where
1038:
785:
651:
487:β
483:α
397:φ
73:β
63:α
30:, the
4044:JSTOR
3951:(PDF)
3924:Notes
3821:from
3784:yards
2182:Mark
1727:find
1559:find
1359:find
1283:find
1106:atan2
1088:is a
1085:atan2
60:angle
3996:ISSN
3829:and
3683:APBC
3655:and
2173:B, C
2164:and
1937:and
1462:>
939:and
493:and
255:and
217:and
203:PACB
130:and
87:and
4036:doi
3986:doi
3865:CPB
3814:CPB
3807:APC
3786:),
3687:iff
3672:ABC
3648:APB
2255:sin
2214:to
2131:tan
1847:cos
1679:cos
1521:sin
1510:sin
1470:sin
1448:sin
1423:sin
1412:sin
1225:tan
1194:tan
1056:sin
1026:sin
887:tan
848:tan
812:tan
773:sin
761:sin
750:sin
738:sin
696:tan
661:tan
633:tan
622:tan
576:tan
539:tan
524:sin
513:sin
455:sin
434:sin
409:tan
369:sin
358:sin
325:sin
314:sin
259:PBC
252:PAC
236:).
137:as
134:CBP
126:as
123:CAP
95:).
58:an
26:In
4092::
4042:.
4032:52
4030:.
4026:.
3994:.
3982:98
3980:.
3976:.
3957:.
3953:.
3855:PC
3849:PB
3843:PA
3835:AB
3789:CB
3779:AC
3643:P'
2202:.
2194:.)
2074:AC
2064:AC
2039:CB
2026:AC
2016:AC
2006:AC
1724:.)
962:BC
960:,
958:AC
265:PC
223:AB
68:CB
52:AC
4050:.
4038::
4002:.
3988::
3959:2
3863:∠
3831:P
3827:C
3819:P
3812:∠
3805:∠
3801:P
3796:C
3794:∠
3752:)
3746:,
3743:2
3740:,
3737:1
3734:=
3731:k
3728:(
3725:,
3719:k
3716:=
3713:C
3710:+
3704:+
3691:C
3663:(
3661:P
3635:P
3600:3
3596:r
3589:3
3585:Q
3579:2
3575:v
3568:=
3561:1
3557:r
3550:2
3546:R
3540:2
3536:v
3529:=
3520:2
3510:P
3507:B
3492:3
3488:r
3481:3
3477:Q
3471:1
3467:v
3460:=
3453:2
3449:r
3442:1
3438:R
3432:1
3428:v
3421:=
3412:2
3402:P
3399:A
3358:3
3354:R
3348:2
3344:R
3340:4
3333:2
3329:)
3323:1
3319:Q
3310:3
3306:R
3302:+
3297:2
3293:R
3289:(
3280:1
3277:=
3268:2
3264:v
3251:3
3247:R
3241:1
3237:R
3233:4
3226:2
3222:)
3216:2
3212:Q
3203:3
3199:R
3195:+
3190:1
3186:R
3182:(
3173:1
3170:=
3161:1
3157:v
3128:0
3124:D
3120:=
3115:2
3111:)
3105:0
3101:C
3092:3
3088:R
3084:(
3076::
3073:3
3070:R
3045:3
3041:r
3036:)
3031:3
3027:Q
3023:,
3018:2
3014:Q
3010:,
3005:1
3001:Q
2997:(
2994:A
2989:2
2985:r
2979:1
2975:r
2968:=
2959:0
2955:D
2942:3
2938:r
2934:2
2929:)
2924:3
2920:r
2914:3
2910:Q
2906:+
2901:2
2897:r
2891:2
2887:Q
2883:+
2878:1
2874:r
2868:1
2864:Q
2860:(
2857:2
2851:)
2846:3
2842:r
2838:+
2833:2
2829:r
2825:+
2820:1
2816:r
2812:(
2809:)
2804:3
2800:Q
2796:+
2791:2
2787:Q
2783:+
2778:1
2774:Q
2770:(
2764:=
2755:0
2751:C
2739:3
2735:r
2728:3
2724:Q
2718:1
2714:r
2707:=
2702:2
2698:R
2689:3
2685:r
2678:3
2674:Q
2668:2
2664:r
2657:=
2648:1
2644:R
2607:2
2597:B
2594:A
2587:=
2578:3
2574:Q
2568:)
2562:+
2556:(
2553:s
2550:=
2541:3
2537:r
2527:2
2517:C
2514:A
2507:=
2498:2
2494:Q
2488:)
2482:(
2479:s
2476:=
2467:2
2463:r
2453:2
2443:C
2440:B
2433:=
2424:1
2420:Q
2413:)
2407:(
2404:s
2401:=
2392:1
2388:r
2375:)
2370:2
2366:z
2362:+
2357:2
2353:y
2349:+
2344:2
2340:x
2336:(
2333:2
2325:2
2321:)
2317:z
2314:+
2311:y
2308:+
2305:x
2302:(
2299:=
2296:)
2293:z
2290:,
2287:y
2284:,
2281:x
2278:(
2275:A
2267:x
2259:2
2251:=
2248:)
2245:x
2242:(
2239:s
2192:P
2188:C
2184:P
2179:.
2168:.
2166:C
2162:A
2158:O
2144:.
2128:2
2119:C
2116:A
2108:=
2099:O
2096:M
2083:O
2079:M
2069:M
2048:P
2031:2
2021:O
1989:P
1969:C
1965:y
1961:,
1956:C
1952:x
1948::
1945:C
1923:A
1919:y
1915:,
1910:A
1906:x
1902::
1899:A
1876:.
1871:)
1868:y
1853:(
1835:C
1832:P
1817:C
1814:B
1805:2
1797:2
1787:C
1784:P
1777:+
1772:2
1762:C
1759:B
1750:=
1741:B
1738:P
1708:.
1703:)
1700:x
1685:(
1667:C
1664:P
1649:C
1646:A
1637:2
1629:2
1619:C
1616:P
1609:+
1604:2
1594:C
1591:A
1582:=
1573:A
1570:P
1516:x
1501:C
1498:A
1484:,
1480:|
1466:|
1458:|
1444:|
1418:y
1403:C
1400:B
1387:{
1382:=
1373:C
1370:P
1345:.
1340:2
1336:W
1330:K
1324:=
1321:y
1315:,
1310:2
1306:W
1303:+
1300:K
1294:=
1291:x
1269:.
1265:]
1261:)
1258:C
1255:+
1249:+
1243:(
1237:2
1234:1
1221:)
1209:4
1200:(
1190:[
1177:2
1174:=
1171:W
1149:.
1146:C
1125:2
1122:=
1119:K
1070:,
1066:)
1047:C
1044:A
1035:,
1017:C
1014:B
1007:(
1001:2
998:n
995:a
992:t
989:a
984:=
945:P
941:y
937:x
921:.
917:)
904:4
894:(
884:)
881:C
878:+
872:+
866:(
860:2
857:1
845:=
842:)
839:y
833:x
830:(
824:2
821:1
788:,
779:y
770:+
767:x
756:y
744:x
732:=
726:)
723:y
720:+
717:x
714:(
708:2
705:1
691:)
688:y
682:x
679:(
673:2
670:1
654:,
645:1
642:+
616:1
610:=
606:)
593:4
583:(
548:.
536:=
530:y
519:x
495:B
491:A
467:.
450:C
447:A
429:C
426:B
418:=
381:.
364:y
349:C
346:B
337:=
320:x
305:C
302:A
293:=
284:C
281:P
257:△
250:△
234:π
229:C
227:∠
219:P
215:C
211:C
207:C
184:C
163:2
160:=
157:y
154:+
151:x
139:y
132:∠
128:x
121:∠
93:P
89:B
85:A
81:C
77:P
44:P
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