X1209–X1285

Saragossa Conjugates, Sharygin Points & Lemoine Homothetic Center

Isogonal and isotomic conjugates of the Saragossa points, Sharygin points, and the Lemoine homothetic center (X1285).
< 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_1209 = ISOGONAL CONJUGATE OF X(1166) (0.5121, 0.1912) X_1217 = CRETAN PERSPECTOR (0.5642, 0.3204) X_1227 = ISOTOMIC CONJUGATE OF X(1168) (0.2551, 0.2499) X_1230 = ISOTOMIC CONJUGATE OF X(1171) (0.4472, 0.1317) X_1236 = ISOTOMIC CONJUGATE OF X(1177) (0.1751, -0.3263) X_1244 = ISOGONAL CONJUGATE OF X(1009) (0.5903, 0.2249) X_1248 = ISOGONAL CONJUGATE OF X(1047) (0.6139, 0.6190) X_1252 = ISOGONAL CONJUGATE OF X(1086) (0.4699, 0.3541) X_1259 = ISOGONAL CONJUGATE OF X(1118) (0.3105, 0.0160) X_1262 = ISOGONAL CONJUGATE OF X(1146) (0.5738, 0.4961) X_1263 = ISOGONAL CONJUGATE OF X(1157) (1.0064, 0.0088) X_1271 = ISOTOMIC CONJUGATE OF X(1132) (-0.0936, -0.4782) X_1278 = CONGRUENT MIDWAY-PARALLELIANS POINT (0.3828, 0.0729) X_1282 = 5th SHARYGIN POINT (-0.1429, 0.8165) X_1283 = 7th SHARYGIN POINT (0.7481, 0.0125) X_1284 = 8th SHARYGIN POINT (0.9846, 0.9156) A B C X3 X1 X1209 X1217 X1227 X1230 X1236 X1244 X1248 X1252 X1259 X1262 X1263 X1271 X1278 X1282 X1283 X1284 (a, b, c) = (5, 6, 7) | AB = 1

X1209–X1285 (Saragossa Conjugates, Sharygin Points & Lemoine Homothetic Center) — 16 representative centers (from 77 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X_1217 — CRETAN PERSPECTOR

X1217CRETAN PERSPECTOR

Using barycentric coordinates, let P =\(p : q : r\) be a point not on a sideline of triangle ABC, and let P' be the isogonal conjugate of P. Let DEF and D'E'F' be the pedal triangles of P and P', respectively. Let

X = PD'∩P'D, Y = PE'∩P'E, Z = PF'∩P'F.

X_1276 — 2nd EVANS PERSPECTOR

X12762nd EVANS PERSPECTOR

Evans conjectured that X1, X484, X1276, X1277 are concyclic, and he reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X1019. (Lawrence Evans, 2/24/2003)
X_1282 — 5th SHARYGIN POINT

X12825th SHARYGIN POINT

X1282 is the perspector of the excentral triangle and the triangle D'E'F' constructed at X1281. Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.
X_1283 — 7th SHARYGIN POINT

X12837th SHARYGIN POINT

X1283 is the center of similitude of the triangles DEF and D'E'F' constructed at X1281. Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.
X_1284 — 8th SHARYGIN POINT

X12848th SHARYGIN POINT

X1284 is the homothetic center of the intouch intriangle and the triangle DEF constructed at X1281. Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.
X_1285 — LEMOINE HOMOTHETIC CENTER

X1285LEMOINE HOMOTHETIC CENTER

X1285 is the homothetic center of the antipedal triangle of X2 and the pedal triangle of X6. (Darij Grinberg, Hyacinthos #6577, 2/21/03). See also X3066.