X1119

2nd HATZIPOLAKIS PERSPECTOR

Let triangle A0B0C0 be as defined for X1118. Let A1 be the orthogonal projection of A0 onto line BC, and define B1 and C1 cyclically. Then triangle A1B1C1 is perspective to ABC, and the perspector is X1119. (Antreas Hatzipolakis, #5321, 4/30/02)
< 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) { 19, 57 }: Central Line L(19, 57) { 34, 269 }: Central Line L(34, 269) { 28, 279 }: Central Line L(28, 279) { 3, 347 }: Central Line L(3, 347) { 142, 281 }: Central Line L(142, 281) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1260 = ISOGONAL CONJUGATE OF X(1119) (0.3557, 0.0365) X_4 = ORTHOCENTER (0.6122, 0.3958) X_7 = GERGONNE POINT (0.5903, 0.2769) X_19 = CLAWSON POINT (0.6246, 0.4244) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_269 = ISOGONAL CONJUGATE OF X(200) (0.6369, 0.3847) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_279 = ISOGONAL CONJUGATE OF X(220) (0.6236, 0.3540) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_347 = ISOTOMIC CONJUGATE OF X(280) (0.4531, -0.0404) X_142 = COMPLEMENT OF X(9) (0.5506, 0.2192) X_281 = -CROSS CONJUGATE OF X(9) (0.5796, 0.3202) X_1119 = 2nd HATZIPOLAKIS PERSPECTOR (0.6322, 0.5039) A B C X1260 X4 X7 X19 X57 X34 X269 X28 X279 X3 X347 X142 X281 X1119 (a, b, c) = (5, 6, 7) | AB = 1

X1119 (2nd HATZIPOLAKIS PERSPECTOR) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (1 - \sec A)/(1 + \cos A)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-0.030578020100\)

X1119 lies on the hyperbola {{A, B, C,X4,X19}}.

▸ Incident Central Lines9 lines

{3, 347}, {4, 7}, {19, 57}, {28, 279}, {34, 269}, {142, 281}, {393, 1086}, {579, 1020}, {915, 934}

▸ Functional Relations4 relations