X1845

ZOSMA TRANSFORM OF X80

X1845 = crosssum of X3 and X1807
< 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) { 34, 46 }: Central Line L(34, 46) { 4, 80 }: Central Line L(4, 80) { 1, 102 }: Central Line L(1, 102) { 36, 186 }: Central Line L(36, 186) { 65, 389 }: Central Line L(65, 389) { 162, 759 }: Central Line L(162, 759) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_4 = ORTHOCENTER (0.6122, 0.3958) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_1 = INCENTER (0.5714, 0.2333) X_102 = Λ(INCENTER, ORTHOCENTER) (0.6364, -0.3897) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_186 = INVERSE-IN-CIRCUMCIRCLE OF X(4) (0.7957, 0.8758) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_389 = CENTER OF THE TAYLOR CIRCLE (0.6078, 0.2800) X_162 = CEVAPOINT OF X(108) AND X(109) (0.4897, 0.6018) X_759 = EVEN (1, 3) CIRCUMCIRCLE POINT (0.9498, -0.1389) X_1845 = ZOSMA TRANSFORM OF X(80) (0.5417, 0.5181) A B C X34 X46 X4 X80 X1 X102 X36 X186 X65 X389 X162 X759 X1845 (a, b, c) = (5, 6, 7) | AB = 1

X1845 (ZOSMA TRANSFORM OF X80) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \((y + z) \sec A : (z + x) \sec B : (x + y) \sec C\), where \(x : y : z\) = X80
Barycentrics: \((y + z) \tan A : (z + x) \tan B : (x + y) \tan C\), where \(x : y : z\) = X80
Search Value (6, 9, 13): \(-0.979433056755\)
▸ Incident Central Lines15 lines

{1, 102}, {4, 80}, {19, 1743}, {34, 46}, {36, 186}, {57, 1795}, {65, 389}, {92, 994}, {117, 1737}, {162, 759}, {407, 1829}, {517, 1361}, {942, 1354}, {1146, 1901}, {1827, 1905}

▸ Functional Relations4 relations

Notes

The following ten points lie on a circle: Xi for i = 11, 36, 65, 80, 108, 759, 1354, 1845, 2588, 2589. (Chris Van Tienhoven, Hyacinthos, January 4, 2011)

Let HaHbHc be the orthic triangle. Let A″ be the A-excenter of AHbHc, and define B″ and C″ cyclically. The lines HaA″, HbB″, HcC″ concur in X1845. (Randy Hutson, January 29, 2018)