X2163

X2-ISOCONJUGATE OF X45

Let JaJbJc be the excentral triangle of a triangle ABC. Let A2 and A3 be the points where the perpendicular bisector of BC meets the internal angle bisector of angle ABC and the internal angle bisector of angle BCA, respectively. Let A'2 and A'3 be the points where the perpendicular bisector of BC meets the external angle bisector of angle ABC and the external angle bisector of angle BCA, respectively. Let Oab, Oac be the circumcenters of triangles JbA2A'3, JcA3A'2, respectively. Define Obc, Oba, Oca, Ocb cyclically. Then the triangle having sidelines OabOac, ObcOba, OcaOcb is perspective to ABC, and the perspector is X2163. See Angel Montesdeoca, HG090218: El centro del triángulo X2163.
< 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) { 6, 36 }: Central Line L(6, 36) { 1, 89 }: Central Line L(1, 89) { 31, 106 }: Central Line L(31, 106) { 57, 1411 }: Central Line L(57, 1411) { 996, 1150 }: Central Line L(996, 1150) { 1126, 1468 }: Central Line L(1126, 1468) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_1 = INCENTER (0.5714, 0.2333) X_89 = ISOGONAL CONJUGATE OF X(45) (0.5963, 0.2697) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_106 = Λ(INCENTER, CENTROID) (1.0000, 0.0000) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_1411 = -BETH CONJUGATE OF X(80) (1.0110, -0.2153) X_996 = INTERSECTION OF LINES X(8)X(58) AND X(10)X(56) (0.5332, 0.2018) X_1150 = INTERSECTION OF LINES X(2)X(6) AND X(3)X(8) (0.4725, 0.1302) X_1126 = ISOGONAL CONJUGATE OF X(1125) (0.5938, 0.2586) X_1468 = -BETH CONJUGATE OF X(958) (0.6033, 0.2751) X_2163 = -ISOCONJUGATE OF X(45) (0.6195, 0.3035) A B C X6 X36 X1 X89 X31 X106 X57 X1411 X996 X1150 X1126 X1468 X2163 (a, b, c) = (5, 6, 7) | AB = 1

X2163 (X2-ISOCONJUGATE OF X45) 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 X45
Barycentrics: \(a^{2}/f(a,b,c) : :\)
Search Value (6, 9, 13): \(0.546201941334\)
▸ Incident Central Lines9 lines

{1, 89}, {3, 2334}, {6, 36}, {31, 106}, {57, 1411}, {996, 1150}, {1055, 2279}, {1126, 1468}, {1220, 1698}

▸ Functional Relations6 relations

Notes

Let A'B'C' be the circumsymedial triangle. Let (OA) be the circle centered at A' and tangent to line BC. Define (OB) and (OC) cyclically. X2163 is the trilinear pole of the Monge line of (OA), (OB), (OC). (Randy Hutson, November 30, 2018)