X3361

INTERSECTION OF LINES X1X_3 AND X7X_1125

Let OA be the circle tangent to side BC at its midpoint and to the circumcircle on the side ob BC opposite A. Define OB and OC cyclically. Let A' be the insimilicanter of OB and OC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X3361. See the reference at X1001.
< 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) { 58, 269 }: Central Line L(58, 269) { 200, 404 }: Central Line L(200, 404) { 7, 1125 }: Central Line L(7, 1125) { 222, 1203 }: Central Line L(222, 1203) { 28, 1435 }: Central Line L(28, 1435) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_200 = -CEVA CONJUGATE OF X(9) (0.4392, 0.1037) X_404 = {X(2),X(3)}-HARMONIC CONJUGATE OF X(21) (0.5538, 0.2429) X_7 = GERGONNE POINT (0.5903, 0.2769) X_1125 = COMPLEMENT OF X(10) (0.5459, 0.2083) X_222 = -CEVA CONJUGATE OF X(56) (0.5653, 0.1568) X_1203 = 3rd SARAGOSSA POINT OF X(58) (0.6034, 0.2712) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_1435 = -BETH CONJUGATE OF X(278) (0.6337, 0.5189) X_3361 = INTERSECTION OF LINES X(1)X(3) AND X(7)X(1125) (0.6211, 0.3246) A B C X1 X3 X58 X269 X200 X404 X7 X1125 X222 X1203 X28 X1435 X3361 (a, b, c) = (5, 6, 7) | AB = 1

X3361 (INTERSECTION OF LINES X1X_3 AND X7X_1125) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((3a + b + c)/(-a + b + c)\)
Search Value (6, 9, 13): \(0.290431015371\)
▸ Incident Central Lines19 lines

{1, 3}, {7, 1125}, {28, 1435}, {34, 2163}, {58, 269}, {200, 404}, {222, 1203}, {223, 1451}, {386, 1458}, {388, 1698}, {610, 2260}, {614, 1448}, {936, 1445}, {961, 1722}, {978, 1400}, {995, 1042}, {1416, 1472}, {1427, 1453}, {1468, 2999}

▸ Functional Relations3 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. A'B'C' is homothetic to the inverse-in-incircle triangle at X3361. (Randy Hutson, July 31 2018)