X971

INTERSECTION OF LINES X3X_9 AND X4X_7

As the isogonal conjugate of a point on the circumcircle, X971 lies on the line at infinity.
< 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) { 4, 7 }: Central Line L(4, 7) { 3, 9 }: Central Line L(3, 9) { 20, 72 }: Central Line L(20, 72) { 5, 142 }: Central Line L(5, 142) { 33, 222 }: Central Line L(33, 222) { 165, 210 }: Central Line L(165, 210) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_7 = GERGONNE POINT (0.5903, 0.2769) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_9 = MITTENPUNKT (0.5110, 0.1615) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_72 = ISOGONAL CONJUGATE OF X(28) (0.4367, 0.0733) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_222 = -CEVA CONJUGATE OF X(56) (0.5653, 0.1568) X_165 = CENTROID OF THE EXCENTRAL TRIANGLE (0.4762, 0.0583) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_971 = INTERSECTION OF LINES X(3)X(9) AND X(4)X(7) (0.6441, 0.7770) A B C X4 X7 X3 X9 X20 X72 X5 X142 X33 X222 X165 X210 X971 (a, b, c) = (5, 6, 7) | AB = 1

X971 (INTERSECTION OF LINES X3X_9 AND X4X_7) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(a^{4}(b + c) - 2a^{3}(b^{2} + c^{2} - bc) + 2a(b - c)^{2}(b^{2} + c^{2} + bc) - (b - c)^{2}(b + c)^{3} : :\)
▸ Incident Central Lines13 lines

{1, 1419}, {3, 9}, {4, 7}, {5, 142}, {6, 990}, {20, 72}, {30, 511}, {33, 222}, {37, 991}, {103, 910}, {165, 210}, {390, 944}, {954, 1012}

▸ Functional Relations8 relations

Notes

X971 is the perspector of triangle ABC and the tangential triangle of the conic that passes through the points A, B, C, X2, and X9. (Randy Hutson, 9/23/2011)