X1824–X1907

Zosma Transforms

Isogonal conjugates of the inverse Mimosa transforms (Z(X)) of classical triangle 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 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_1824 = ZOSMA TRANSFORM OF X(2) (0.6198, 0.4097) X_1830 = ZOSMA TRANSFORM OF X(11) (0.5714, 0.4666) X_1833 = ZOSMA TRANSFORM OF X(16) (0.7191, 0.7639) X_1834 = ZOSMA TRANSFORM OF X(28) (0.5987, 0.2534) X_1835 = ZOSMA TRANSFORM OF X(30) (0.6564, 0.6795) X_1836 = ZOSMA TRANSFORM OF X(33) (0.6066, 0.3254) X_1845 = ZOSMA TRANSFORM OF X(80) (0.5417, 0.5181) X_1849 = ZOSMA TRANSFORM OF X(175) (0.6306, 0.4953) X_1853 = ZOSMA TRANSFORM OF X(204) (0.5941, 0.7953) X_1854 = ZOSMA TRANSFORM OF X(208) (0.5955, 0.5353) X_1861 = ZOSMA TRANSFORM OF X(238) (0.8061, 0.8443) X_1878 = ZOSMA TRANSFORM OF X(519) (0.7415, 0.6332) X_1879 = ZOSMA TRANSFORM OF X(563) (0.5328, 0.3490) X_1886 = ZOSMA TRANSFORM OF X(910) (0.6597, 0.7902) X_1897 = ZOSMA TRANSFORM OF X(1054) (0.4360, 0.5756) X_1898 = ZOSMA TRANSFORM OF X(1068) (0.5903, 0.1207) A B C X3 X1 X1824 X1830 X1833 X1834 X1835 X1836 X1845 X1849 X1853 X1854 X1861 X1878 X1879 X1886 X1897 X1898 (a, b, c) = (5, 6, 7) | AB = 1

X1824–X1907 (Zosma Transforms) — 16 representative centers (from 84 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Zosma Transforms 1824-1907 (Zozma transforms)

The Zosma transform of a point X =\(x : y : z\) is the isogonal conjugate of the inverse Mimosa transform of X, given by trilinears
(y + z) sec A : (z + x) sec B : (x + y)sec C.

(Zosma is another star name.)

X_1904 — ZOSMA TRANSFORM OF X_1468

X1904ZOSMA TRANSFORM OF X1468

As a point on the Euler line, X1904 has Shinagawa coefficients \((FS^{2}, -(E - F)S^{2} - 2\$bc\$S^{2} + 2\$bcS_{B}S_{C}\$)\).