X3601

INTERSECTION OF LINES X1X_3 AND X9X_21

X3601, the {X1, X3611}-harmonic conjugate of X3576, was found by Peter Moses in connection with the 1st Liu point.
< 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, 3611 }: Central Line L(1, 3611) { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 9, 21 }: Central Line L(9, 21) { 28, 33 }: Central Line L(28, 33) { 20, 226 }: Central Line L(20, 226) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3576 = INTERSECTION OF LINES X(1)X(3) AND X(2)X(515) (0.5238, 0.1458) X_1 = INCENTER (0.5714, 0.2333) X_3611 = 2nd AYME-MOSES PERSPECTOR (0.5837, 0.1426) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_9 = MITTENPUNKT (0.5110, 0.1615) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_3601 = INTERSECTION OF LINES X(1)X(3) AND X(9)X(21) (0.5521, 0.1977) A B C X3576 X1 X3611 X3 X9 X21 X28 X33 X20 X226 X3601 (a, b, c) = (5, 6, 7) | AB = 1

X3601 (INTERSECTION OF LINES X1X_3 AND X9X_21) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(3 \cos A + \cos B + \cos C + 1 : :\)
Search Value (6, 9, 13): \(2.555200973956\)
▸ Incident Central Lines37 lines

{1, 3}, {2, 950}, {5, 3586}, {7, 3522}, {8, 3158}, {9, 21}, {10, 3486}, {20, 226}, {28, 33}, {37, 610}, {58, 212}, {73, 991}, {77, 738}, {84, 943}, {100, 1706}, {142, 390}, {200, 958}, {376, 3487}, {386, 1453}, {405, 936}, {443, 497}, {452, 3452}, {515, 3085}, {516, 3485}, {579, 1449}, {631, 1210}, {728, 2329}, {938, 3523}, {1012, 1490}, {1055, 3100}, {1104, 2999}, {1193, 2293}, {1253, 1468}, {1439, 3532}, {1682, 3056}, {1698, 1837}, {1876, 3516}

▸ Functional Relations2 relations

  • X3601 = {X1,X3}-harmonic conjugate of X57

  • X3601 = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles

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 intouch triangle at X3601. (Randy Hutson, July 31 2018)