X1001–X1069

Classical Midpoints, Euler Intercepts & Cevapoints

Midpoints of classical centers, Euler line intercepts, collinearity intersections, cevapoints, and pure-angle trilinear centers.
< 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 nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line 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_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_1001 = MIDPOINT OF X(1) AND X(9) (0.5412, 0.1974) X_1004 = INTERCEPT OF EULER LINE AND LINE X(7)X(100) (0.5701, 0.2854) X_1007 = INTERCEPT OF LINES X(2)X(6) AND X(4)X(99) (0.7128, 0.3884) X_1015 = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.7982, 0.1348) X_1016 = -CROSS CONJUGATE OF X(99) (0.3832, 0.2666) X_1017 = POINT ALFIRK (0.4054, 0.3972) X_1018 = -CROSS CONJUGATE OF X(1) (0.1051, 0.4358) X_1020 = POINT ALGEDI (0.4031, 0.5979) X_1022 = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514) (1.0000, -0.0000) X_1025 = INTERSECTION OF LINES X(2)X(7) AND X(100)X(109) (1.0996, 1.0180) X_1028 = POINT ALGENIB (0.4600, 0.1238) X_1032 = CYCLOCEVIAN CONJUGATE OF X(20) (0.3836, -0.1007) X_1053 = POINT ALHENA (0.7302, -0.1555) X_1054 = 6th SHARYGIN POINT (1.4286, -0.2333) X_1055 = POINT ALIOTH (1.0459, 1.3483) X_1062 = INTERSECTION OF LINES X(1)X(3) AND X(5)X(33) (0.4633, 0.0347) A B C X3 X1 X5 X2 X1001 X1004 X1007 X1015 X1016 X1017 X1018 X1020 X1022 X1025 X1028 X1032 X1053 X1054 X1055 X1062 (a, b, c) = (5, 6, 7) | AB = 1

X1001–X1069 (Classical Midpoints, Euler Intercepts & Cevapoints) — 16 representative centers (from 69 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_1001 — MIDPOINT OF X_1 AND X_9

X1001MIDPOINT OF X1 AND X9

Let A* be the tangency point of OA and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of OB and OC. The lines A*A**, B*B**, C*C** concur in X1001.
X_1015 — EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

X1015EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

The circle having center X39 and radius 2R \(\sin^{2}\)ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X115, and its internal and external centers of similitude with the incircle are X1500 and X1015, respectively. (based on notes from Peter J. C. Moses, 5/29/03)
X_1056 — POINT ALCYONE

X1056POINT ALCYONE

X1056 = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X4