X2148–X2254

Isotomic Conjugates (X2-Isoconjugates) I

Isotomic conjugates (X2-isoconjugates) of triangle centers X5 through X892.
< 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_2148 = -ISOCONJUGATE OF X(5) (0.6157, 0.2500) X_2149 = -ISOCONJUGATE OF X(11) (0.5572, 0.4660) X_2152 = -ISOCONJUGATE OF X(14) (1.1000, 1.3310) X_2154 = -ISOCONJUGATE OF X(16) (0.9495, -0.1417) X_2155 = -ISOCONJUGATE OF X(20) (0.5380, 1.2070) X_2156 = -ISOCONJUGATE OF X(22) (0.5755, 0.9221) X_2157 = -ISOCONJUGATE OF X(23) (0.8254, -0.2862) X_2159 = -ISOCONJUGATE OF X(30) (0.8357, -0.4858) X_2160 = -ISOCONJUGATE OF X(35) (0.6185, 0.3356) X_2162 = -ISOCONJUGATE OF X(43) (0.5193, 0.2016) X_2164 = -ISOCONJUGATE OF X(46) (0.1231, 0.0273) X_2165 = -ISOCONJUGATE OF X(47) (-0.1110, -0.0295) X_2169 = -ISOCONJUGATE OF X(53) (0.5228, 0.0936) X_2170 = -ISOCONJUGATE OF X(59) (0.7601, 0.0792) X_2172 = -ISOCONJUGATE OF X(66) (0.4601, -0.1665) X_2178 = -ISOCONJUGATE OF X(90) (0.6843, 0.4716) A B C X3 X1 X2148 X2149 X2152 X2154 X2155 X2156 X2157 X2159 X2160 X2162 X2164 X2165 X2169 X2170 X2172 X2178 (a, b, c) = (5, 6, 7) | AB = 1

X2148–X2254 (Isotomic Conjugates (X2-Isoconjugates) I) — 16 representative centers (from 107 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_2163 — X_2-ISOCONJUGATE OF X_45

X2163-ISOCONJUGATE OF X45

Let JaJbJc be the excentral triangle of a triangle ABC. Let A2 and A3 be the points where the perpendicular bisector of BC meets the internal angle bisector of angle ABC and the internal angle bisector of angle BCA, respectively. Let A'2 and A'3 be the points where the perpendicular bisector of BC meets the external angle bisector of angle ABC and the external angle bisector of angle BCA, respectively. Let Oab, Oac be the circumcenters of triangles JbA2A'3, JcA3A'2, respectively. Define Obc, Oba, Oca, Ocb cyclically. Then the triangle having sidelines OabOac, ObcOba, OcaOcb is perspective to ABC, and the perspector is X2163. See Angel Montesdeoca, HG090218: El centro del triángulo X2163.
X_2245 — X_2-ISOCONJUGATE OF X_759

X2245-ISOCONJUGATE OF X759

Let L denote the line having trilinears of X2245 as coefficients. Then L is the line passing through X1 perpendicular to the Euler line.