X3624

H(X10; M, 1/2)

X3624 = homothetic center of ABC and cross-triangle of Aquila and anti-Aquila triangles
< 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, 2 }: Nagel Line { 40, 140 }: Central Line L(40, 140) { 57, 191 }: Central Line L(57, 191) { 36, 405 }: Central Line L(36, 405) { 34, 451 }: Central Line L(34, 451) { 35, 474 }: Central Line L(35, 474) 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_2 = CENTROID (0.5374, 0.2000) X_40 = BEVAN POINT (0.4286, -0.0292) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_405 = EULER LINE INTERCEPT OF LINE X(1)X(6) (0.5325, 0.1870) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_451 = -HIRST INVERSE OF X(424) (0.5621, 0.2645) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_474 = EULER LINE INTERCEPT OF LINE X(10)X(56) (0.5441, 0.2176) X_3624 = H(X(10); M, 1/2) (0.5423, 0.2047) A B C X1 X2 X40 X140 X57 X191 X36 X405 X34 X451 X35 X474 X3624 (a, b, c) = (5, 6, 7) | AB = 1

X3624 (H(X10; M, 1/2)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(bc(3a +2 b + 2c) : :\)
Barycentrics: \(3a + 2b + 2c : 3b + 2c + 2a : 3c + 2a + 2b\)
Search Value (6, 9, 13): \(2.495217323484\)
▸ Alternate Coordinates2 forms

  • Trilinears: \(bc + rR : ca + rR : ab + rR\)

  • Trilinears: \(r + 4 R \sin B \sin C : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines30 lines

{1, 2}, {3, 1699}, {5, 3576}, {9, 583}, {11, 3601}, {12, 1420}, {34, 451}, {35, 474}, {36, 405}, {40, 140}, {57, 191}, {58, 748}, {63, 3337}, {90, 3255}, {165, 631}, {226, 3361}, {377, 3583}, {442, 3586}, {443, 1479}, {515, 3090}, {516, 3523}, {517, 3526}, {595, 750}, {940, 1203}, {1213, 1449}, {1385, 1656}, {1706, 3035}, {2478, 3585}, {3306, 3336}, {3339, 3485}

▸ 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 3rd Euler triangle at X3624. (Randy Hutson, July 31 2018)