X3585

INTERSECTION OF LINES X1X_4 AND X12X_30

X3585 = reflection of X35 in X12
< 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) { 35, 12 }: Central Line L(35, 12) { 1, 4 }: Incenter–Orthocenter Line { 5, 36 }: Central Line L(5, 36) { 12, 30 }: Central Line L(12, 30) { 65, 79 }: Central Line L(65, 79) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) X_3585 = INTERSECTION OF LINES X(1)X(4) AND X(12)X(30) (0.5909, 0.3109) A B C X35 X12 X1 X4 X5 X36 X30 X65 X79 X3585 (a, b, c) = (5, 6, 7) | AB = 1

X3585 (INTERSECTION OF LINES X1X_4 AND X12X_30) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

  • Central Lines: {35, 12}

  • Related Centers: X35 ({X), X12 ({X)

Trilinears: \(bc(b^{4} + c^{4} - a^{4} - 2b^{2}c^{2} - a^{2}bc)\)
Search Value (6, 9, 13): \(-0.616361680878\)
▸ Linear Combinations1 combination
▸ Incident Central Lines23 lines

{1, 4}, {5, 36}, {10, 191}, {11, 546}, {12, 30}, {13, 2307}, {20, 498}, {55, 382}, {56, 381}, {65, 79}, {115, 172}, {149, 3244}, {316, 1909}, {355, 1836}, {377, 1698}, {535, 2975}, {993, 2476}, {1254, 1725}, {1539, 3024}, {1781, 1826}, {2964, 3072}, {3070, 3301}, {3071, 3299}

▸ Functional Relations4 relations

  • X3585 = reflection of X35 in X12

  • X3585 = homothetic center of 2nd isogonal triangle of X1 and Johnson triangle; see X36

  • X3585 = {X1,X4}-harmonic conjugate of X3583

  • X3585 = homothetic center of Ehrmann vertex-triangle and anti-tangential midarc triangle