X1105

CEVAPOINT OF X3 AND X4

X1105 is the cevapoint of X3 and X4.
< 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) { 20, 393 }: Central Line L(20, 393) { 225, 412 }: Central Line L(225, 412) { 243, 411 }: Central Line L(243, 411) { 4, 801 }: Central Line L(4, 801) { 185, 648 }: Central Line L(185, 648) { 3, 1093 }: Central Line L(3, 1093) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_393 = -CROSS CONJUGATE OF X(4) (0.6247, 0.5263) X_225 = -CEVA CONJUGATE OF X(65) (0.6246, 0.4451) X_412 = EULER X(29)-2nd-SUBSTITUTION POINT (0.6482, 0.4899) X_243 = -LINE CONJUGATE OF X(73) (0.7516, 0.9505) X_411 = EULER X(21)-2nd-SUBSTITUTION POINT (0.4414, -0.0514) X_801 = ISOGONAL CONJUGATE OF X(800) (0.4263, 0.1232) X_185 = NAGEL POINT OF THE ORTHIC TRIANGLE (0.6033, 0.1643) X_648 = TRILINEAR POLE OF EULER LINE (0.4563, 0.5855) X_1093 = TRILINEAR CUBE OF X(4) (0.6189, 0.5667) X_1105 = CEVAPOINT OF X(3) AND X(4) (0.5529, 0.3087) A B C X3 X4 X20 X393 X225 X412 X243 X411 X801 X185 X648 X1093 X1105 (a, b, c) = (5, 6, 7) | AB = 1

X1105 (CEVAPOINT OF X3 AND X4) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec A)/(\cos^{2} B + \cos^{2} C)\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A) f(A,B,C)\)
Search Value (6, 9, 13): \(-309.394308647941\)
▸ Incident Central Lines7 lines

{3, 1093}, {4, 801}, {20, 393}, {185, 648}, {225, 412}, {243, 411}, {378, 847}

▸ Functional Relations3 relations

Notes

Let A'B'C' be the cevian triangle of X3. Let LA be the reflection of the line B'C' in the line BC, and define LB and LC cyclically. Let A″ = LB∩LC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1105. (Randy Hutson, 9/23/2011)