X79

ISOGONAL CONJUGATE OF X35

A construction for X79 appears in Dasari Naga Vijay Krishna, On the Feuerbach Triangle.
< 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) { 191, 442 }: Central Line L(191, 442) { 1, 30 }: Central Line L(1, 30) { 9, 46 }: Central Line L(9, 46) { 21, 36 }: Central Line L(21, 36) { 65, 80 }: Central Line L(65, 80) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_442 = COMPLEMENT OF SCHIFFLER POINT (0.5418, 0.2114) X_1 = INCENTER (0.5714, 0.2333) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_9 = MITTENPUNKT (0.5110, 0.1615) X_46 = -CEVA CONJUGATE OF X(1) (0.6917, 0.4543) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_65 = ORTHOCENTER OF THE INTOUCH TRIANGLE (0.6041, 0.2933) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_79 = ISOGONAL CONJUGATE OF X(35) (0.5976, 0.3019) A B C X191 X442 X1 X30 X9 X46 X21 X36 X65 X80 X79 (a, b, c) = (5, 6, 7) | AB = 1

X79 (ISOGONAL CONJUGATE OF X35) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/(1 + 2 \cos A) : :\)
Barycentrics: \(1/(b^{2} + c^{2} - a^{2} + bc) : :\)
Search Value (6, 9, 13): \(0.046914305978\)
▸ Alternate Coordinates4 forms

  • Trilinears: \(bc/(b^{2} + c^{2} - a^{2} + bc)\)

  • Trilinears: \((\sin A/2)(\sin 3B/2)(\sin 3C/2) : :\)

  • Trilinears: \(\sin(A/2) \csc(3A/2) : :\)

  • Barycentrics: \(1/(b c + 2 S_{A}) : :\)

▸ Linear Combinations1 combination
▸ Incident Central Lines14 lines

{1, 30}, {2, 3647}, {8, 758}, {9, 46}, {12, 484}, {21, 36}, {33, 1063}, {34, 1061}, {35, 226}, {57, 90}, {65, 80}, {104, 946}, {314, 320}, {388, 1000}

▸ Functional Relations14 relations

Notes

Let A' be the reflection of X1 in sideline BC, and define B' and C' cyclically. Then the lines AA', BB', CC' concur in X79. (Eric Danneels, Hyacinthos 7892, 9/13/03)

A'B'C' is also the reflection triangle of X1. The lines AA', BB', CC' concur in X79. (Randy Hutson, July 20, 2016)

Let P and Q be the intersections of line BC and circle {X1,2r}. Let X = X1. Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X79. (Compare to X592, where the circle is the 1st Lemoine circle) (Randy Hutson, July 20, 2016)

Let A25B25C25 be Gemini triangle 25. Let A' be the perspector of conic {A, B, C,B25,C25}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X79. (Randy Hutson, January 15, 2019)