X1393–X1477

Beth Conjugates & Harmonic Conjugates

Fixed-point P-beth conjugates of P and harmonic conjugates along central lines.
< 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_1393 = -BETH CONJUGATE OF X(5) (0.6196, 0.3615) X_1395 = -BETH CONJUGATE OF X(25) (0.6371, 0.5131) X_1399 = -BETH CONJUGATE OF X(35) (0.6178, 0.2762) X_1404 = -BETH CONJUGATE OF X(44) (0.5590, 0.1678) X_1411 = -BETH CONJUGATE OF X(80) (1.0110, -0.2153) X_1413 = -BETH CONJUGATE OF X(84) (0.6037, 0.6378) X_1414 = -BETH CONJUGATE OF X(99) (0.3368, 0.5766) X_1415 = -BETH CONJUGATE OF X(101) (0.4138, 0.6170) X_1417 = -BETH CONJUGATE OF X(106) (1.0000, 0.0000) X_1424 = -BETH CONJUGATE OF X(194) (0.6912, 0.4122) X_1429 = -BETH CONJUGATE OF X(239) (0.9040, 0.8443) X_1433 = -BETH CONJUGATE OF X(271) (0.6277, 0.9396) X_1438 = -BETH CONJUGATE OF X(294) (0.9961, -0.1135) X_1443 = -BETH CONJUGATE OF X(320) (0.9441, 1.0954) X_1444 = -BETH CONJUGATE OF X(332) (0.4762, 0.0921) X_1447 = -BETH CONJUGATE OF X(350) (1.1014, 1.0207) A B C X3 X1 X1393 X1395 X1399 X1404 X1411 X1413 X1414 X1415 X1417 X1424 X1429 X1433 X1438 X1443 X1444 X1447 (a, b, c) = (5, 6, 7) | AB = 1

X1393–X1477 (Beth Conjugates & Harmonic Conjugates) — 16 representative centers (from 85 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Beth Conjugates 1393-1477

If X is a triangle center then X = P-beth conjugate of P for some triangle center P. Using the definition of beth conjugate (in the Glossary), it is easy to prove that the P-beth conjugate of P is the trilinear product P*X56,
so that P = X*X8. In trilinears,

if P =\(p : q : r\), then (P-beth conjugate of P) =\(up : vq : wr\), where

\(u : v : w\) = a/(b + c - a) : b/(c + a - b) : c/(a + b - c), or equivalently,
\(u : v : w\) = 1 - cos A : 1 - cos B : 1 - cos C.

Following is a list of pairs (i,j) for which Xi = Xj-beth conjugate of Xj:

1,8 2,312 3,78 4,318 6,9 7,75 8,341 9,346 12,1089 19,281 21,1043 25,33 28,29 31,55 32,41 34,4 41,220 42,210 48,219 55,200 56,1 57,2 58,21 59,765 60,1098 63,345 65,10 73,72 77,69 78,1265 81,333 84,280 85,76 86,314 101,644 109,100 110,643 142,1229 174,556 181,756 184,212 190,646 212,1260 220,728 221,40 222,63 223,329 226,321 244,11 255,1259 266,188 269,7 273,264 278,92 279,85 326,1264 347,322 348,304 479,1088 513,522 552,873 603,3 604,6 604,6 608,19 614,497 649,650 651,190 662,645 664,668 667,663 738,279 757,261 849,60 934,664 951,1257 961,1220 1014,86 1027,885 1042,65 1104,950 1106,56 1118,158 1119,273 1193,960 1214,306 1253,480 1254,12 1284,740

X_1471 — X_1001-BETH CONJUGATE OF X_1001

X1471-BETH CONJUGATE OF X1001

Let A1B1C1 and A2B2C2 be the 1st and 2nd Auriga triangles. Let A' be the trilinear product A1*A2, and define B', C' cyclically. The lines AA', BB', CC' concur in X1471. (Randy Hutson, March 21, 2019)