X92

CEVAPOINT OF INCENTER AND CLAWSON POINT

Let LA be the line through X4 parallel to the internal bisector of angle A, and let

  • A' = BC∩LA. Define B' and C' cyclically.

< 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) { 4, 8 }: Fuhrmann Line { 1, 29 }: Central Line L(1, 29) { 19, 27 }: Central Line L(19, 27) { 47, 91 }: Central Line L(47, 91) { 31, 162 }: Central Line L(31, 162) { 7, 189 }: Central Line L(7, 189) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_19 = CLAWSON POINT (0.6246, 0.4244) X_4 = ORTHOCENTER (0.6122, 0.3958) X_8 = NAGEL POINT (0.4694, 0.1333) X_29 = CEVAPOINT OF INCENTER AND ORTHOCENTER (0.5903, 0.3384) X_27 = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER (0.6182, 0.4112) X_47 = -BETH CONJUGATE OF X(34) (0.7279, 0.4389) X_91 = ISOGONAL CONJUGATE OF X(47) (-0.0865, -0.0205) X_31 = 2nd POWER POINT (0.6228, 0.3008) X_162 = CEVAPOINT OF X(108) AND X(109) (0.4897, 0.6018) X_7 = GERGONNE POINT (0.5903, 0.2769) X_189 = CYCLOCEVIAN CONJUGATE OF X(8) (0.6131, 0.7055) X_92 = CEVAPOINT OF INCENTER AND CLAWSON POINT (0.5949, 0.3639) A B C X1 X19 X4 X8 X29 X27 X47 X91 X31 X162 X7 X189 X92 (a, b, c) = (5, 6, 7) | AB = 1

X92 (CEVAPOINT OF INCENTER AND CLAWSON POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\csc 2A : \csc 2B : \csc 2C\)
Barycentrics: \(\sec A : \sec B : \sec C\)
Search Value (6, 9, 13): \(31.410531526384\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cot A + \tan A : :\)

▸ Linear Combinations2 combinations

  • X92 = \(2r(r+2R)\) X4 - \(((r + 2R)^{2} - s^{2})\) X8

  • X92 = \((\sec A)\) A + \((\sec B)\) B + \((\sec C)\) C, where A, B, C denote angles and A, B, C denote vertices

▸ Incident Central Lines71 lines

{1, 29}, {2, 273}, {4, 8}, {7, 189}, {10, 1838}, {19, 27}, {25, 242}, {28, 2975}, {31, 162}, {33, 1897}, {34, 1220}, {38, 240}, {40, 412}, {47, 91}, {48, 2167}, {53, 4415}, {55, 243}, {56, 1940}, {57, 653}, {81, 2995}, {85, 331}, {100, 917}, {108, 1311}, {171, 1430}, {226, 342}, {239, 607}, {255, 1087}, {257, 297}, {264, 306}, {304, 561}, {345, 3262}, {388, 1118}, {394, 1943}, {406, 1068}, {427, 2969}, {429, 3948}, {429, 3948}, {459, 1446}, {497, 1857}, {518, 1859}, {608, 894}, {651, 2988}, {823, 2349}, {938, 3176}, {942, 1148}, {960, 1882}, {984, 1860}, {994, 1845}, {1146, 1952}, {1172, 2997}, {1211, 1865}, {1309, 2717}, {1435, 3306}, {1585, 1659}, {1621, 4183}, {1707, 1733}, {1726, 1746}, {1731, 1751}, {1785, 4656}, {1842, 1891}, {1844, 3874}, {1870, 5136}, {1947, 2994}, {1954, 1955}, {1956, 2632}, {1973, 3112}, {2064, 3596}, {2331, 5256}, {2399, 4391}, {3064, 4468}, {4198, 4968}

▸ Functional Relations32 relations

Notes

Alexei Myakishev, “The M-Configuration of a Triangle,” Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA', BB', CC' concur in X92. He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A″ = A1A2∩BC. Define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X92.

Suppose that T = A'B'C' is a central triangle. Let A″ be the pole with respect to the polar circle of the line B'C', and define B″ and C″ cyclically. The appearance of T in the following list means that the lines AA″, BB″, CC″ concur in X92: Feurerbach, incentral, excentral, extangents, Apollonius, mixtilinear excentral. (Randy Hutson, December 26, 2015)