X1903

ZOSMA TRANSFORM OF X1249

X1903 = crosspoint of X84 and X189
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 3, 9 }: Central Line L(3, 9) { 6, 33 }: Central Line L(6, 33) { 19, 64 }: Central Line L(19, 64) { 37, 73 }: Central Line L(37, 73) { 69, 189 }: Central Line L(69, 189) { 71, 210 }: Central Line L(71, 210) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_84 = ISOGONAL CONJUGATE OF X(40) (0.6071, 0.6816) X_189 = CYCLOCEVIAN CONJUGATE OF X(8) (0.6131, 0.7055) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_9 = MITTENPUNKT (0.5110, 0.1615) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_19 = CLAWSON POINT (0.6246, 0.4244) X_64 = ISOGONAL CONJUGATE OF X(20) (0.5579, 1.5943) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_73 = CROSSPOINT OF INCENTER AND CIRCUMCENTER (0.5487, 0.1428) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_71 = ISOGONAL CONJUGATE OF X(27) (0.4807, 0.0915) X_210 = -CEVA CONJUGATE OF X(37) (0.4925, 0.1466) X_1903 = ZOSMA TRANSFORM OF X(1249) (0.6093, 0.6932) A B C X84 X189 X3 X9 X6 X33 X19 X64 X37 X73 X69 X71 X210 X1903 (a, b, c) = (5, 6, 7) | AB = 1

X1903 (ZOSMA TRANSFORM OF X1249) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((y + z) \sec A : :\), where \(x : y : z\) = X1249
Barycentrics: \(a (b + c)/(a^{3} + a^{2} (b + c) - a (b + c)^{2} - (b - c)^{2} (b + c)) : :\)
Search Value (6, 9, 13): \(14.691454578722\)
▸ Incident Central Lines11 lines

{3, 9}, {6, 33}, {19, 64}, {37, 73}, {65, 1826}, {69, 189}, {71, 210}, {226, 1439}, {285, 1798}, {478, 1413}, {1419, 1422}

▸ Functional Relations3 relations

Notes

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X478) at Bc and Ca, and define Tb, Tc cyclically. Let Ta' be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb', Tc' cyclically. Let Va = TbTb'∩TcTc', Vb = TcTc'∩TaTa', Vc = TaTa'∩TbTb'. The lines AVa, BVb, CVc concur in X1903. (See also X65.) (Randy Hutson, July 20, 2016)