X476

TIXIER POINT

The reflection of X110 in the Euler line; X476 is on the circumcircle. (Michel Tixier, 5/9/98). Also, X476 is the center of the polar conic of X30 with respect to the Neuberg cubic; this conic is a rectangular hyperbola passing through the incenter, the excenters, and X30. (Peter Yff, 5/23/99)
< 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) { 30, 74 }: Central Line L(30, 74) { 23, 94 }: Central Line L(23, 94) { 111, 230 }: Central Line L(111, 230) { 3, 477 }: Central Line L(3, 477) { 110, 523 }: Central Line L(110, 523) { 2, 842 }: Central Line L(2, 842) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_526 = ISOGONAL CONJUGATE OF X(476) (1.0501, 0.4856) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_74 = ISOGONAL CONJUGATE OF EULER INFINITY POINT (0.7484, -0.3437) X_23 = FAR-OUT POINT (1.3871, 2.4232) X_94 = ISOGONAL CONJUGATE OF X(50) (0.9069, -0.1331) X_111 = PARRY POINT (0.9885, -0.0457) X_230 = -CEVA CONJUGATE OF X(114) (0.9746, 0.6698) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_477 = TIXIER ANTIPODE (0.0176, -0.0644) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_523 = ISOGONAL CONJUGATE OF X(110) (1.0856, -0.0096) X_2 = CENTROID (0.5374, 0.2000) X_842 = ISOGONAL CONJUGATE OF X(542) (-0.0101, 0.1156) X_476 = TIXIER POINT (0.9824, 0.2685) A B C X526 X30 X74 X23 X94 X111 X230 X3 X477 X110 X523 X2 X842 X476 (a, b, c) = (5, 6, 7) | AB = 1

X476 (TIXIER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/[(1 + 2 \cos 2A) \sin(B - C)] : :\)
Barycentrics: \(1/{(b^{2} - c^{2})[(a^{2} - b^{2} - c^{2})^{2} - b^{2}c^{2}]} : :\)
Search Value (6, 9, 13): \(-0.026518972092\)

X476 lies on the Hutson-Parry circle.

▸ Incident Central Lines127 lines

{2, 842}, {3, 477}, {4, 16221}, {5, 14979}, {13, 2379}, {14, 2378}, {20, 2693}, {21, 2687}, {22, 2697}, {23, 94}, {24, 15112}, {30, 74}, {99, 850}, {100, 4036}, {101, 4024}, {102, 7424}, {103, 5196}, {104, 1325}, {105, 7469}, {106, 7478}, {107, 7480}, {109, 13486}, {110, 523}, {111, 230}, {112, 2501}, {125, 17511}, {143, 15907}, {146, 1553}, {183, 2857}, {186, 1300}, {250, 933}, {328, 2373}, {376, 841}, {403, 1299}, {468, 3563}, {511, 9161}, {512, 9160}, {524, 9184}, {526, 16170}, {549, 11749}, {655, 15439}, {691, 4226}, {729, 11060}, {755, 14902}, {759, 2166}, {804, 20404}, {843, 6792}, {858, 1297}, {915, 2074}, {917, 2073}, {930, 23181}, {935, 4230}, {953, 3109}, {1141, 2070}, {1290, 3658}, {1292, 7475}, {1294, 2071}, {1296, 7472}, {1298, 19167}, {1302, 1632}, {1304, 4240}, {1316, 2698}, {1379, 13722}, {1380, 13636}, {1494, 9141}, {1637, 23588}, {1648, 14846}, {1789, 2716}, {1995, 2453}, {2367, 20573}, {2395, 2715}, {2407, 10420}, {2409, 10423}, {2410, 9060}, {2452, 11422}, {2688, 4184}, {2689, 7450}, {2690, 4243}, {2691, 4236}, {2694, 16049}, {2695, 4225}, {2696, 11634}, {2706, 2979}, {2752, 4228}, {2758, 7419}, {2766, 4246}, {2799, 17708}, {2858, 23342}, {2966, 9979}, {3060, 16978}, {3153, 18401}, {3154, 15059}, {3260, 15295}, {3448, 6070}, {4235, 10098}, {4238, 10101}, {4608, 6578}, {5468, 10425}, {5663, 16169}, {5897, 16386}, {5899, 13597}, {5994, 20579}, {5995, 20578}, {6032, 9831}, {6325, 11594}, {6644, 15111}, {6757, 11101}, {6795, 15080}, {7575, 13530}, {8599, 11636}, {8705, 11593}, {9033, 15395}, {9070, 15455}, {9202, 14185}, {9203, 14187}, {10296, 18300}, {10412, 15329}, {10989, 14388}, {12077, 23357}, {12113, 18318}, {14934, 15035}, {15061, 16340}, {15168, 18359}, {15646, 20480}, {16089, 18817}, {18323, 18576}, {18403, 22751}

▸ Functional Relations14 relations

Notes

Let La, Lb, Lc be the lines through A, B, C, respectively parallel to the Euler line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, respectively. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X476; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 18, 2015)

Let La, Lb, Lc be the lines through A, B, C, respectively parallel to the orthic axis. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, respectively. Let A' = Mb∩Mc, and define B' and C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. The centroid of A'B'C' is X476; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, November 18, 2015)

X476 is the center of the polar conic of X30 with respect to the Neuberg cubic, which passes through the following points: incenter, excenters, X30, X5, X523, and the excentral-isogonal conjugates of X5663 and X7724. (Randy Hutson, November 18, 2015)