X3243

REFLECTION OF X9 IN X1

X3243 is the reflection of X9 in X1.
< 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) { 9, 1 }: Incenter–Symmedian Line { 1, 6 }: Incenter–Symmedian Line { 7, 145 }: Central Line L(7, 145) { 57, 100 }: Central Line L(57, 100) { 390, 527 }: Central Line L(390, 527) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1 = INCENTER (0.5714, 0.2333) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_7 = GERGONNE POINT (0.5903, 0.2769) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_390 = REFLECTION OF GERGONNE POINT IN INCENTER (0.5526, 0.1897) X_527 = DIRECTION OF VECTOR AX + BX + CX, where X = X(7) (0.8698, 0.6836) X_3243 = REFLECTION OF X(9) IN X(1) (0.6319, 0.3051) A B C X9 X1 X6 X7 X145 X57 X100 X390 X527 X3243 (a, b, c) = (5, 6, 7) | AB = 1

X3243 (REFLECTION OF X9 IN X1) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{2} + 3b^{2} + 3c^{2} - 2bc - 4ca - 4ab : :\)
Search Value (6, 9, 13): \(-0.191355680316\)
▸ Linear Combinations1 combination
▸ Incident Central Lines9 lines

{1, 6}, {7, 145}, {57, 100}, {65, 2136}, {390, 527}, {516, 944}, {942, 1706}, {971, 1482}, {1056, 2550}

▸ Functional Relations2 relations

Notes

Let A'B'C' be the reflection of ABC in X1. ABC and A'B'C' intersect at 6 points, which lie on an ellipse centered at X1 with perspector X10390. Let A″ be the intersection of the tangents to this ellipse at the points where it intersects BC, and define B″ and C″ cyclically. (i.e., A″B″C″ is the polar triangle of the ellipse.) A'B'C' and A″B″C″ are perspective at X3243. (Randy Hutson, August 29, 2018)