X279

ISOGONAL CONJUGATE OF X220

X279 is the isogonal conjugate of X220.
< 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) { 1, 7 }: Soddy Line { 2, 85 }: Central Line L(2, 85) { 56, 105 }: Central Line L(56, 105) { 144, 220 }: Central Line L(144, 220) { 57, 479 }: Central Line L(57, 479) { 304, 346 }: Central Line L(304, 346) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_220 = -CEVA CONJUGATE OF X(55) (0.4826, 0.1278) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_2 = CENTROID (0.5374, 0.2000) X_85 = ISOTOMIC CONJUGATE OF X(9) (0.5610, 0.2421) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_105 = Λ(INCENTER, SYMMEDIAN POINT) (0.9714, -0.0933) X_144 = ANTICOMPLEMENT OF X(7) (0.4317, 0.0461) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_479 = -CROSS CONJUGATE OF X(279) (0.6392, 0.4221) X_304 = ISOTOMIC CONJUGATE OF X(19) (0.3683, 0.0513) X_346 = ISOTOMIC CONJUGATE OF X(279) (0.3948, 0.0827) X_279 = ISOGONAL CONJUGATE OF X(220) (0.6236, 0.3540) A B C X220 X1 X7 X2 X85 X56 X105 X144 X57 X479 X304 X346 X279 (a, b, c) = (5, 6, 7) | AB = 1

X279 (ISOGONAL CONJUGATE OF X220) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\csc A \tan^{2} A/2 : \csc B \tan^{2} B/2 : \csc C \tan^{2} C/2\)
Barycentrics: \(\tan^{2} A/2 : \tan^{2} B/2 : \tan^{2} C/2\)
Search Value (6, 9, 13): \(0.116757051176\)
▸ Alternate Coordinates1 form

  • Barycentrics: \((a + b - c)^{2} (a - b + c)^{2} : :\)

▸ Linear Combinations1 combination

  • X279 = \(2 s^{2}\) X1 - \((r + 4 R)^{2}\) X7

X279 lies on the circumconic {A, B, C, X1, X2}}, the cubics K041, K360, K605, K623, K1011, K1044, K1189.

▸ Incident Central Lines181 lines

{1, 7}, {2, 85}, {4, 1565}, {6, 1170}, {8, 7273}, {10, 31994}, {11, 15511}, {28, 1014}, {56, 105}, {57, 479}, {65, 1002}, {69, 1257}, {75, 1219}, {76, 32017}, {81, 1407}, {88, 658}, {100, 2377}, {109, 2369}, {144, 220}, {145, 664}, {165, 3599}, {196, 37378}, {218, 651}, {222, 2982}, {226, 5308}, {253, 7219}, {273, 1440}, {274, 16713}, {277, 8732}, {278, 1847}, {280, 17880}, {291, 1254}, {294, 4209}, {304, 346}, {307, 5232}, {329, 25930}, {330, 7209}, {331, 13149}, {345, 21605}, {388, 1390}, {394, 9965}, {519, 25718}, {553, 39948}, {555, 16015}, {604, 28079}, {644, 28981}, {668, 6552}, {764, 43930}, {938, 34059}, {942, 955}, {959, 24471}, {961, 6046}, {985, 1106}, {1068, 38461}, {1086, 4626}, {1111, 3086}, {1146, 10405}, {1214, 8813}, {1224, 19855}, {1255, 6354}, {1275, 5376}, {1319, 24796}, {1400, 39970}, {1406, 38877}, {1418, 4000}, {1432, 41777}, {1439, 18732}, {1445, 16572}, {1447, 5265}, {1461, 2224}, {1462, 16502}, {1617, 40154}, {1788, 43037}, {1997, 33780}, {2006, 43047}, {2340, 36854}, {2355, 42382}, {2401, 24002}, {2724, 24016}, {2898, 5274}, {3008, 5435}, {3091, 17181}, {3146, 4872}, {3210, 39696}, {3218, 7183}, {3227, 4569}, {3241, 25716}, {3323, 18343}, {3361, 10521}, {3474, 30623}, {3476, 24797}, {3485, 4059}, {3501, 6168}, {3616, 32086}, {3617, 33298}, {3623, 32007}, {3673, 14986}, {3676, 21132}, {3752, 44794}, {3767, 4403}, {3875, 9797}, {3911, 39963}, {3926, 20924}, {4099, 4552}, {4188, 6516}, {4452, 6553}, {4454, 40862}, {4488, 10029}, {4554, 18135}, {4617, 34051}, {4644, 6610}, {5226, 29571}, {5252, 24798}, {5261, 7179}, {5281, 9446}, {5526, 41563}, {5905, 34401}, {6063, 30710}, {6356, 37179}, {6603, 20059}, {6743, 32099}, {6904, 41826}, {6995, 39732}, {7080, 30806}, {7132, 7175}, {7147, 43071}, {7181, 7288}, {7182, 20911}, {7270, 10513}, {7371, 10490}, {7674, 8271}, {8232, 16601}, {8809, 10429}, {9305, 28071}, {9316, 9441}, {9442, 43750}, {9445, 31526}, {10025, 26658}, {10529, 38468}, {10939, 31391}, {11019, 31527}, {11433, 20211}, {12447, 17272}, {14878, 40293}, {15634, 18328}, {16705, 37870}, {17080, 37597}, {17084, 30571}, {17213, 28074}, {17863, 33673}, {18624, 40940}, {18625, 37887}, {19605, 32446}, {19789, 35058}, {20070, 38866}, {20111, 40868}, {20247, 35312}, {20925, 32834}, {23978, 44190}, {23983, 34403}, {24181, 30379}, {24858, 41803}, {25723, 38314}, {26125, 32009}, {27253, 40779}, {30545, 39703}, {30699, 39694}, {30719, 37626}, {32020, 46406}, {32624, 36152}, {32836, 33939}, {32841, 32851}, {33935, 40892}, {34018, 36838}, {34578, 37771}, {35094, 35158}, {36079, 41905}, {36603, 36621}, {37579, 38900}, {37758, 40014}, {43928, 43932}

▸ Functional Relations36 relations

Notes

X279 is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Gergonne line. The center of the inellipse is X4000. (Randy Hutson, October 15, 2018)

Let A1B1C1 and A2B2C2 be the 1st and 2nd Conway triangles. Let A' be the barycentric product A1*A2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X279. (Randy Hutson, July 11, 2019)

An exarc circle is a circle tangent to two sides of a triangle ABC and externally tangent to the circumcircle of ABC. The point whose distance to the sides BC, CA, AB are proportional to the respective radii of the exarc circles is X279. The point having distances to the sides proportional to the radii of the inarc circles is X7. See Martin Lukarevski, “Exarc radii and the Finsler-Hadwiger inequality”, The Mathematical Gazette 106, issue 565, March 2022, pp. 138-143.