X1312–X1353

Moses Intersections, Circumcircle Inverses, Wernau Points & Similicenters

Moses orthocentroidal-Euler intersections, circumcircle inverses, 1st and 2nd Wernau points (X1337, X1338), and similicenters of the circumcircle, nine-point circle, orthocentroidal circle, and Brocard circle.
< 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_1312 = 1st MOSES INTERSECTION (0.6472, 0.4872) X_1315 = 4th MOSES INTERSECTION (0.4699, 0.0233) X_1316 = 5th MOSES INTERSECTION (0.5678, 0.2795) X_1317 = REFLECTION OF X(11) IN X(1) (0.4082, 0.3999) X_1318 = ISOGONAL CONJUGATE OF X(1317) (1.0000, 0.0000) X_1319 = BEVAN-SCHRÖDER POINT (0.9524, 0.9331) X_1321 = 1st YIU SQUARES PERSPECTOR (0.6050, 0.3673) X_1322 = 2nd YIU SQUARES PERSPECTOR (0.0183, 0.0422) X_1323 = FLETCHER POINT (1.0262, 1.2854) X_1326 = INVERSE-IN-CIRCUMCIRCLE OF X(58) (1.1309, 1.1440) X_1328 = ARCTAN(-3) KIEPERT POINT (0.4997, 1.5024) X_1331 = ORTHOCORRESPONDENT OF X(101) (-0.2308, 0.2512) X_1334 = POINT ALYA (0.5324, 0.1760) X_1338 = 2nd WERNAU POINT (1.0969, 0.1421) X_1339 = NAGEL-SCHRÖDER POINT (-0.2381, -0.2333) X_1350 = REFLECTION OF X(6) IN X(3) (0.4000, -0.0631) A B C X3 X1 X1312 X1315 X1316 X1317 X1318 X1319 X1321 X1322 X1323 X1326 X1328 X1331 X1334 X1338 X1339 X1350 (a, b, c) = (5, 6, 7) | AB = 1

X1312–X1353 (Moses Intersections, Circumcircle Inverses, Wernau Points & Similicenters) — 16 representative centers (from 42 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_1314 — 3rd MOSES INTERSECTION

X13143rd MOSES INTERSECTION

X1314 is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).
X_1315 — 4th MOSES INTERSECTION

X13154th MOSES INTERSECTION

X1315 is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).
X_1321 — 1st YIU SQUARES PERSPECTOR

X13211st YIU SQUARES PERSPECTOR

Let \(BB_{A}\)CAC be the external square on side BC, and define \(CC_{B}\)ABA and \(AA_{C}\)BCB cyclically. Let X = BCB∩CBC and X' = BAB∩CAC, and define Y, Z and Y', Z' cyclically. The lines AX, BY, CZ concur in X4, and the lines AX', BY', CZ' concur in X485. The lines XX', YY', ZZ' concur in X1321, as shown in
X_1323 — FLETCHER POINT

X1323FLETCHER POINT

X1323 is the point of intersection of the line X1X7 and the trilinear polar of X7. These two lines are orthogonal.
X_1327 — ARCTAN(3) KIEPERT POINT

X1327ARCTAN(3) KIEPERT POINT

Let \(BB_{A}\)CAC be the external square on side BC, and define \(CC_{B}\)ABA and \(AA_{C}\)BCB cyclically. The lines ABBA, BCCB, CAAC form a triangle perspective to triangle ABC, and the perspector is X1327.
X_1338 — 2nd WERNAU POINT

X13382nd WERNAU POINT

Let A'BC be the internal equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X1338. For details, see X1337.
X_1339 — NAGEL-SCHRÖDER POINT

X1339NAGEL-SCHRÖDER POINT

Let NaNbNc be the extouch triangle of ABC; viz., Na is where the A-excircle meets line BC, and NaNbNc is the pedal triangle of X40. Let I = incenter of ABC. The circles (AIX’), (BIY’), (CIZ’) concur in two points: I and X1339. (Jean-Pierre Ehrmann, Hyacinthos #6545)