X1118

1st HATZIPOLAKIS PERSPECTOR

Let TaTbTc be the intouch triangle of ABC. Let CA be the point other than Tc in which the perpendicular to BC from Tc meets the incircle, let BA be the point other than Tb in which the perpendicular to BC from Tb meets the incircle, and let A0 be the point of intersection of lines BCA and CBA. Define B0 and C0 cyclically. Then triangle A0B0C0 is perspective to ABC, and the perspector is X1118. (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, 65 }: Central Line L(4, 65) { 28, 56 }: Central Line L(28, 56) { 24, 108 }: Central Line L(24, 108) { 19, 208 }: Central Line L(19, 208) { 34, 207 }: Central Line L(34, 207) { 20, 243 }: Central Line L(20, 243) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1259 = ISOGONAL CONJUGATE OF X(1118) (0.3105, 0.0160) X_4 = ORTHOCENTER (0.6122, 0.3958) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_24 = PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC TRIANGLE (0.6627, 0.5278) X_108 = Ψ(CIRCUMCENTER, INCENTER) (0.5320, 0.6114) X_19 = CLAWSON POINT (0.6246, 0.4244) X_208 = -CEVA CONJUGATE OF X(34) (0.6242, -0.2728) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_207 = -CEVA CONJUGATE OF X(34) (0.5818, 0.9198) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_243 = -LINE CONJUGATE OF X(73) (0.7516, 0.9505) X_1118 = 1st HATZIPOLAKIS PERSPECTOR (0.6247, 0.5521) A B C X1259 X4 X65 X28 X56 X24 X108 X19 X208 X34 X207 X20 X243 X1118 (a, b, c) = (5, 6, 7) | AB = 1

X1118 (1st 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 \(f(A,B,C) = (1 - \cos A)/\cos^{2} A\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.055842400756\)

X1118 lies on the hyperbola {{A, B, C,X4,X19}}.

▸ Incident Central Lines9 lines

{4, 65}, {7, 286}, {12, 281}, {19, 208}, {20, 243}, {24, 108}, {28, 56}, {34, 207}, {92, 388}

▸ Functional Relations4 relations

Notes

X1118 is the trilinear product A0*B0*C0, where A0, B0, C0 are as defined above. (Randy Hutson, January 15, 2019)