X2166

X2-ISOCONJUGATE OF X50

X2166 = trilinear pole of line X661X1953
< 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) { 10, 94 }: Central Line L(10, 94) { 65, 79 }: Central Line L(65, 79) { 1, 564 }: Central Line L(1, 564) { 476, 759 }: Central Line L(476, 759) { 37, 1989 }: Central Line L(37, 1989) { 897, 1733 }: Central Line L(897, 1733) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_1 = INCENTER (0.5714, 0.2333) X_564 = INTERSECTION OF LINES X(1)X(1048) AND X(47,91) (0.4993, 0.3100) X_476 = TIXIER POINT (0.9824, 0.2685) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9498, -0.1389) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_1989 = ISOGONAL CONJUGATE OF X(323) (0.9916, -0.2095) X_897 = ISOGONAL CONJUGATE OF X(896) (0.9765, -0.0384) X_1733 = MIMOSA TRANSFORM OF X(98) (1.1555, 0.7565) X_2166 = -ISOCONJUGATE OF X(50) (0.9476, -0.1665) A B C X10 X94 X65 X79 X1 X564 X476 X759 X37 X1989 X897 X1733 X2166 (a, b, c) = (5, 6, 7) | AB = 1

X2166 (X2-ISOCONJUGATE OF X50) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\), where f(a,b,c) is as in X50
Barycentrics: \(a^{2}/f(a,b,c) : b^{2}/f(b,c,a): c^{2}/f(c,a,b)\)
Search Value (6, 9, 13): \(-0.118447913451\)
▸ Alternate Coordinates3 forms

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = \sin A \csc 3A\)

  • Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A, B, C) = 1/(1 + 2 \cos 2A)\)

  • Trilinears: \(h(A,B,C) : h(B,C,A) : h(C,A,B)\), where \(h(A, B, C) = 1/(1 - 4 \cos^{2} A)\)

▸ Incident Central Lines8 lines

{1, 564}, {10, 94}, {37, 1989}, {65, 79}, {162, 2190}, {476, 759}, {897, 1733}, {1141, 2222}

▸ Functional Relations6 relations

Notes

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. X2166 is the trilinear product A1*A2 = B1*B2 = C1*C2. (Randy Hutson, June 27, 2018)

Let VaVbVc be the Ehrmann vertex-triangle. Let A' be the trilinear product Vb*Vc, and define B', C' cyclically. The lines AA', BB', CC' concur in X2166. (Randy Hutson, June 27, 2018)

Let VaVbVc and SaSbSc be the Ehrmann vertex-triangle and Ehrmann side-triangle, resp. Let A' be the trilinear product Va*Sa, and define B', C' cyclically. The lines AA', BB', CC' concur in X2166. (Randy Hutson, June 27, 2018)