X1420

X145-BETH CONJUGATE OF X145

X1420 = homothetic center of 1st Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 34, 106 }: Central Line L(34, 106) { 84, 104 }: Central Line L(84, 104) { 9, 604 }: Central Line L(9, 604) { 226, 452 }: Central Line L(226, 452) { 73, 995 }: Central Line L(73, 995) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_84 = ISOGONAL CONJUGATE OF X(40) (0.6071, 0.6816) X_104 = ANTIPODE OF X(100) (0.8571, -0.2624) X_9 = MITTENPUNKT (0.5110, 0.1615) X_604 = -CEVA CONJUGATE OF X(31) (0.6442, 0.3744) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_452 = -HIRST INVERSE OF X(449) (0.5268, 0.1722) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_995 = MIDPOINT OF X(1) AND X(43) (0.6036, 0.2648) X_1420 = -BETH CONJUGATE OF X(145) (0.6753, 0.4242) A B C X145 X1 X3 X34 X106 X84 X104 X9 X604 X226 X452 X73 X995 X1420 (a, b, c) = (5, 6, 7) | AB = 1

X1420 (X145-BETH CONJUGATE OF X145) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((1 - \cos A) u(a,b,c)\), where \(u :v : w\) = X145
Search Value (6, 9, 13): \(-0.073173528548\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(3 \cos A + \cos B + \cos C - 3 : \cos A + 3 \cos B + \cos C - 3 : \cos A + \cos B + 3 \cos C - 3\) (Peter Moses, March 12, 2011)

  • Trilinears: \(-1 + \sin(A/2)\csc(B/2)\csc(C/2)\) (Randy Hutson, August 23, 2011)

  • Trilinears: \((3a - b - c)/(a - b - c) : :\) (Randy Hutson, July 31, 2018)

▸ Incident Central Lines22 lines

{1, 3}, {9, 604}, {21, 1412}, {34, 106}, {73, 995}, {84, 104}, {109, 1106}, {222, 1191}, {223, 1104}, {226, 452}, {269, 1279}, {386, 1450}, {388, 1125}, {595, 603}, {610, 1108}, {738, 934}, {936, 956}, {944, 1210}, {1042, 1149}, {1201, 1419}, {1394, 1457}, {1400, 1449}

▸ Functional Relations6 relations

Notes

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically; A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. The triangle A'B'C' is homothetic to the Hutson intouch triangle at X1420. (Randy Hutson, July 31 2018)