X844–X900

Hatzipolakis-Yiu Centers, Hirst Inverses & Crossdifferences

Hatzipolakis-Yiu points, the Yiu angle point (X848), Kiepert hyperbola multipliers, Hirst inverses, and Euler line intercepts.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_844 = INTERSECTION OF LINES X(166)X(167) AND X(173)X(503) (0.4606, 0.0024) X_845 = ∩X(173)X(503) (0.4168, -0.0880) X_846 = 4th SHARYGIN POINT (0.5437, 0.1969) X_847 = -CROSS CONJUGATE OF X(4) (-0.1806, -0.0821) X_849 = 4th HATZIPOLAKIS-YIU POINT (0.6376, 0.3351) X_852 = -LINE CONJUGATE OF X(4) (0.3028, -0.4138) X_854 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(56) (0.7792, 0.8325) X_855 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(57) (0.8408, 0.9939) X_856 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(63) (0.3538, -0.2804) X_862 = INTERCEPT OF EULER LINE AND TRILINEAR POLAR OF X(19) (0.7262, 0.6939) X_871 = ISOTOMIC CONJUGATE OF X(869) (0.4515, 0.1398) X_875 = ISOGONAL CONJUGATE OF X(874) (0.9992, -0.0257) X_877 = INTERSECTION OF LINES X(4)X(69) AND X(99)X(112) (0.8267, 0.7499) X_878 = ISOGONAL CONJUGATE OF X(877) (1.1192, 0.0431) X_886 = INTERSECTION OF LINES X(99)X(669) AND X(512)X(670) (0.7371, -0.1549) X_889 = INTERSECTION OF LINES X(99)X(898) AND X(190)X(649) (0.8340, -0.1151) A B C X3 X1 X844 X845 X846 X847 X849 X852 X854 X855 X856 X862 X871 X875 X877 X878 X886 X889 (a, b, c) = (5, 6, 7) | AB = 1

X844–X900 (Hatzipolakis-Yiu Centers, Hirst Inverses & Crossdifferences) — 16 representative centers (from 57 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_846 — 4th SHARYGIN POINT

X8464th SHARYGIN POINT

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3001, June 11, 2001. For the construction as a Sharygin point, see the description at X1281.
X_849 — 4th HATZIPOLAKIS-YIU POINT

X8494th HATZIPOLAKIS-YIU POINT

Let D denote the circumcircle of triangle ABC. Let DA be the circle tangent to sideline BC and tangent to D at A. Let BA = AC∩DA and CA = AB∩DA, and define CB, CB and AC, AB cyclically. Define A' = CBAB∩ACBC, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to ABC, and X849 is the center of homothety. See A. Hatzipolakis and P. Yiu, Hyacinthos #2056-2070, December, 2000.
X_877 — INTERSECTION OF LINES X_4X_69 AND X_99X_112

X877INTERSECTION OF LINES X4X_69 AND X99X_112

Let T1 be the tangential triangle of hyperbola {A, B, C,X99,PU(37)}}. Let T2 be the tangential triangle of hyperbola {A, B, C,X4,X112,PU(39)}}. Let A'B'C' be the vertex triangle of T1 and T2. The lines AA', BB', CC' concur in X877. (Randy Hutson, December 29, 2015)