X₁₅₅

EIGENCENTER OF ORTHIC TRIANGLE

X₁₅₅ = X₄-of-tangential-triangle

5, 6, 7 3, 4, 5 Kepler Kimberling

X₁₅₅ (EIGENCENTER OF ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 6}, {3, 49}

• Related Centers: X₅ (NINE-POINT CENTER), X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE)

Trilinears: \((\cos A)[\cos^{2} B + \cos^{2} C - \cos^{2} A] : :\)

Barycentrics: \(a^{2} S_{A} [a^{2} (S_{A}^{2} - S_{B} S_{C}) - S_{A} (S_{B}^{2} + S_{C}^{2})] : :\)

Search Value (6, 9, 13): \(-0.264320420299\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2} (a^{2} - b^{2} - c^{2}) (a^{6} - 3 a^{4} (b^{2} + c^{2}) + a^{2} (3 b^{4} - 2 b^{2} c^{2} + 3 c^{4}) - (b^{2} - c^{2})^{2} (b^{2} + c^{2})) : :\)

▸ Linear Combinations2 combinations

• X₁₅₅ = \(4 R^{2}\) X₅ - \(S_{\omega}\) X₆, \((J^{2} - 3)\) X₃ - \(2(J^{2} - 2)\) X₄₉, \((J^{2} + 1)\) X₂₄ - \(2 J^{2}\) X₁₁₀, \((J^{2} + 3)\) X₂₅ - \(4\) X₅₂

• 3 X₂ - 4 X₉₈₂₀,X₃ - 3 X₃₁₆₇,3 X₃ - 2 X₇₆₈₉,3 X₃ - 4 X₁₂₀₃₈,X₃ + 2 X₁₅₀₈₃,2 X₅ - 3 X₅₆₅₄,2 X₅ + X₉₉₃₆,4 X₅ - 3 X₁₄₈₅₂,3 X₆ - 2 X₈₅₄₈,2 X₂₆ - 3 X₁₅₄,3 X₅₁ - 2 X₁₂₂₃₅,3 X₅₁ - X₂₁₆₅₁,X₆₈ - 3 X₅₆₅₄,2 X₆₈ - 3 X₁₄₈₅₂,4 X₁₁₀ - X₁₅₀₈₅,3 X₁₅₄ - 4 X₁₅₆,3 X₁₅₄ - X₁₇₈₃₄,4 X₁₅₆ - X₁₇₈₃₄,3 X₃₈₁ - 4 X₅₄₄₈,3 X₃₈₁ - 2 X₉₉₂₇,3 X₃₈₁ - X₁₂₄₂₉,2 X₁₁₄₇ - 3 X₃₁₆₇,3 X₁₁₄₇ - X₇₆₈₉,3 X₁₁₄₇ - 2 X₁₂₀₃₈,4 X₁₁₄₇ - X₁₂₁₆₃,2 X₁₁₄₇ + X₁₂₁₆₄,X₁₄₉₈ + 2 X₁₆₂₆₆,5 X₁₆₅₆ - 4 X₅₄₄₉,4 X₁₆₅₈ - 5 X₁₇₈₂₁,3 X₁₈₅₃ - 4 X₁₃₃₇₁,3 X₁₈₅₃ - 2 X₃₂₁₄₀,3 X₃₀₆₀ + X₁₂₂₇₁,9 X₃₁₆₇ - 2 X₇₆₈₉,9 X₃₁₆₇ - 4 X₁₂₀₃₈,6 X₃₁₆₇ - X₁₂₁₆₃,3 X₃₁₆₇ + X₁₂₁₆₄,3 X₃₁₆₇ + 2 X₁₅₀₈₃,5 X₃₅₆₇ - X₁₂₂₈₂,9 X₅₀₅₄ - 8 X₂₀₁₉₁,3 X₅₀₉₃ - X₆₃₉₁,4 X₅₄₄₈ - X₁₂₄₂₉,3 X₅₅₈₇ - X₉₈₉₆,3 X₅₆₅₄ + X₉₉₃₆,3 X₅₈₈₆ - 2 X₁₂₂₅₉,2 X₆₁₉₃ + X₁₂₂₉₃,X₆₁₉₃ + 2 X₂₂₆₆₀,4 X₇₆₈₉ - 3 X₁₂₁₆₃,2 X₇₆₈₉ + 3 X₁₂₁₆₄,X₇₆₈₉ + 3 X₁₅₀₈₃,X₈₅₄₈ - 3 X₁₉₁₃₉,5 X₈₅₆₇ - 4 X₃₂₁₃₈,4 X₉₈₂₀ - X₁₁₄₁₁,3 X₉₉₀₉ - 5 X₁₄₅₃₀,2 X₉₉₂₅ + X₁₁₄₇₇,2 X₉₉₃₆ + 3 X₁₄₈₅₂,4 X₁₀₂₈₂ - 3 X₁₄₀₇₀,3 X₁₀₆₀₆ - 4 X₁₁₂₅₀,3 X₁₁₂₀₆ - X₃₁₃₀₅,4 X₁₁₂₅₅ - 3 X₁₇₈₁₃,8 X₁₂₀₃₈ - 3 X₁₂₁₆₃,4 X₁₂₀₃₈ + 3 X₁₂₁₆₄,2 X₁₂₀₃₈ + 3 X₁₅₀₈₃,X₁₂₁₆₃ + 2 X₁₂₁₆₄,X₁₂₁₆₃ + 4 X₁₅₀₈₃,X₁₂₂₉₃ - 4 X₂₂₆₆₀,2 X₁₂₈₉₃ - 3 X₃₂₆₀₉,4 X₁₃₅₆₁ - 5 X₃₁₂₈₃,3 X₁₈₃₂₄ - 4 X₃₂₁₇₁,4 X₁₈₃₇₇ - 3 X₁₈₄₀₅,5 X₁₉₁₃₂ - 4 X₁₉₁₅₄

X₁₅₅ lies on the following curves: Feuerbach hyperbola of T, Jerabek hyperbola of T, Kiepert circumhyperbola of T, where T = tangential triangle; the Feuerbach circumhyperbola of the orthic triangle; the cubics K006, K044, K176, K339, K364, K389, K500, K742;.

▸ Incident Central Lines298 lines

{1, 90}, {2, 7592}, {3, 49}, {4, 254}, {5, 6}, {11, 10071}, {12, 10055}, {19, 12417}, {20, 323}, {22, 1614}, {24, 110}, {25, 52}, {26, 154}, {30, 1498}, {33, 9931}, {34, 19471}, {51, 7529}, {54, 7503}, {55, 6237}, {56, 7352}, {64, 2935}, {69, 3547}, {81, 6824}, {113, 19504}, {125, 19360}, {136, 20422}, {140, 17811}, {143, 13861}, {159, 511}, {182, 7393}, {193, 3089}, {195, 381}, {215, 9672}, {219, 26921}, {221, 14988}, {222, 24467}, {225, 8757}, {265, 15317}, {275, 19196}, {287, 28695}, {343, 3549}, {355, 12422}, {371, 8909}, {372, 8943}, {378, 9938}, {382, 399}, {389, 6642}, {403, 2904}, {427, 23307}, {450, 1075}, {454, 6503}, {517, 9928}, {524, 16252}, {525, 8151}, {542, 8549}, {550, 16936}, {568, 7506}, {569, 5891}, {576, 8681}, {578, 5907}, {610, 2323}, {631, 15032}, {648, 1093}, {651, 1068}, {858, 11457}, {916, 8053}, {940, 6862}, {952, 9933}, {1060, 19349}, {1062, 19354}, {1160, 8904}, {1161, 8903}, {1199, 3090}, {1209, 32341}, {1350, 2916}, {1351, 1598}, {1368, 18914}, {1478, 18970}, {1479, 12428}, {1493, 15060}, {1495, 9714}, {1503, 14790}, {1593, 11472}, {1594, 11442}, {1596, 13142}, {1611, 10011}, {1616, 19907}, {1620, 15646}, {1625, 2207}, {1656, 5449}, {1658, 17821}, {1740, 3072}, {1829, 7497}, {1839, 7534}, {1853, 13371}, {1899, 11585}, {1974, 19141}, {1994, 3091}, {1995, 3567}, {2063, 18913}, {2072, 25738}, {2192, 8144}, {2477, 9659}, {2914, 11271}, {2929, 6102}, {2930, 5609}, {2979, 10323}, {2990, 23697}, {3047, 12219}, {3053, 32661}, {3060, 6152}, {3098, 15606}, {3197, 8141}, {3216, 6911}, {3289, 23115}, {3435, 10692}, {3448, 23306}, {3511, 19597}, {3517, 8780}, {3518, 16880}, {3527, 5093}, {3542, 6515}, {3546, 18909}, {3548, 11064}, {3580, 7505}, {3627, 15811}, {3628, 17825}, {3629, 15873}, {3832, 11004}, {3851, 14627}, {4383, 6959}, {4846, 31829}, {5012, 7509}, {5020, 5462}, {5050, 10170}, {5054, 20191}, {5070, 15037}, {5072, 15038}, {5085, 7516}, {5095, 16534}, {5374, 31386}, {5398, 16471}, {5412, 12424}, {5413, 12425}, {5453, 6914}, {5587, 9896}, {5706, 6917}, {5876, 7526}, {5886, 12259}, {5890, 17928}, {5898, 12310}, {5972, 12227}, {6000, 12085}, {6090, 9730}, {6146, 12420}, {6212, 19216}, {6213, 19215}, {6241, 11413}, {6243, 7517}, {6623, 32605}, {6643, 6776}, {6756, 31802}, {6800, 7512}, {6804, 14912}, {6944, 32911}, {7074, 32141}, {7391, 16659}, {7401, 14826}, {7404, 11427}, {7484, 13336}, {7485, 7999}, {7486, 15018}, {7487, 15741}, {7488, 9544}, {7507, 18474}, {7514, 11591}, {7527, 9716}, {7552, 12325}, {7553, 31383}, {7723, 19457}, {8200, 12415}, {8207, 12416}, {8220, 12426}, {8221, 12427}, {8227, 16472}, {8251, 19350}, {8276, 12239}, {8277, 12240}, {8538, 10602}, {8541, 9926}, {8567, 32138}, {8718, 33524}, {8954, 10962}, {8976, 13909}, {9019, 15581}, {9545, 14118}, {9645, 10535}, {9705, 11464}, {9715, 26864}, {9815, 10127}, {9909, 14530}, {9923, 9996}, {9967, 19459}, {10112, 18390}, {10224, 18356}, {10255, 32539}, {10257, 26937}, {10282, 14070}, {10575, 12174}, {10606, 11250}, {10620, 12901}, {10625, 11414}, {10628, 12412}, {10634, 19363}, {10635, 19364}, {10641, 10659}, {10642, 10660}, {10796, 12193}, {10897, 19355}, {10898, 19356}, {10942, 12430}, {10943, 12431}, {11206, 31305}, {11249, 23361}, {11255, 17813}, {11265, 17819}, {11266, 17820}, {11267, 17826}, {11268, 17827}, {11403, 16194}, {11422, 13434}, {11424, 15030}, {11426, 11479}, {11433, 18934}, {11449, 32534}, {11499, 12328}, {11557, 22550}, {11560, 12168}, {11562, 12165}, {11819, 31815}, {11898, 19362}, {12082, 23061}, {12112, 33703}, {12233, 18420}, {12290, 14094}, {12315, 14915}, {12319, 14683}, {12358, 13198}, {12362, 31804}, {12601, 19461}, {12602, 19462}, {12603, 19463}, {12604, 19464}, {12605, 19467}, {12606, 19468}, {12825, 15463}, {12893, 32609}, {12902, 19479}, {13039, 19465}, {13040, 19466}, {13321, 18369}, {13561, 31283}, {13567, 18951}, {13951, 13970}, {14128, 32136}, {14216, 23335}, {14480, 15112}, {15033, 15058}, {15106, 15115}, {15750, 32110}, {15800, 32332}, {15818, 31807}, {16618, 16789}, {17849, 32428}, {17975, 20764}, {18324, 32171}, {18377, 18405}, {18396, 18404}, {18534, 26883}, {19125, 19131}, {19132, 19154}, {19170, 19179}, {19180, 19210}, {19189, 19194}, {19358, 19428}, {19359, 19429}, {19446, 19486}, {19447, 19487}, {19460, 22834}, {22120, 22146}, {22808, 22953}, {23070, 23144}, {23294, 30744}, {26913, 31282}, {26920, 26922}, {26944, 30771}, {32153, 34046}, {32251, 32275}

▸ Functional Relations30 relations

• X₁₅₅ = X₄-of-tangential-triangle. This point is also the center of the circle which cuts (extended) lines BC, CA, AB in pairs of points A' and A″, B' and B″, C' and C″, respectively, such that angles A'AA″, B'BB″, C'CC″ are all right angles. This is the Dou circle, described in Jordi Dou, Problem 1140, Crux Mathematicorum, 28 (2002) 461-462.

• X₁₅₅ = midpoint of X_i and X_j for these {i,j}: {3, 12164}, {4, 6193}, {68, 9936}, {1147, 15083}, {1351, 19588}, {6237, 6238}, {12160, 12166}, {12319, 14683}, {12422, 12423}, {16266, 32139}, {17838, 17847}

• X₁₅₅ = reflection of X_i in X_j for these {i,j}: {3, 1147}, {4, 22660}, {6, 19139}, {26, 156}, {64, 12084}, {68, 5}, {1498, 32139}, {2931, 110}, {3448, 23306}, {7387, 6759}, {7689, 12038}, {9927, 5448}, {10620, 12901}, {11411, 12359}, {12085, 13346}, {12163, 3}, {12164, 15083}, {12293, 4}, {12302, 5504}, {12359, 9820}, {12421, 13292}, {12429, 9927}, {12902, 19479}, {14216, 23335}, {14852, 5654}, {15085, 2931}, {15136, 3292}, {16003, 15115}, {17834, 26}, {18356, 10224}, {19458, 12161}, {21651, 12235}, {32140, 13371}

• X₁₅₅ = isogonal conjugate of X₂₅₄

• X₁₅₅ = isotomic conjugate of the polar conjugate of X₁₆₀₉

• X₁₅₅ = X₄-of-tangential-triangle

• X₁₅₅ = complement of X₁₁₄₁₁

• X₁₅₅ = anticomplement of X₁₂₃₅₉

• X₁₅₅ = circumcircle-inverse of X₁₂₀₉₅

• X₁₅₅ = polar-circle inverse of X₁₆₁₇₂

• X₁₅₅ = isogonal conjugate of the polar conjugate of X₆₅₁₅

• X₁₅₅ = polar conjugate of the isotomic conjugate of X₆₅₀₃

• X₁₅₅ = excentral-isogonal conjugate of X₂₉₆₀

• X₁₅₅ = tangential-isogonal conjugate of X₂₆

• X₁₅₅ = orthic-isogonal conjugate of X₃

• X₁₅₅ = X₁₅₅ = X_i-Ceva conjugate of X_j for these (i,j): {4, 3}, {1993, 6}, {3193, 1}, {3542, 454}, {6193, 9937}, {6515, 1609}, {11441, 1498}, {12134, 2918}, {14516, 2917}, {15242, 18126}

• X₁₅₅ = X_i-isoconjugate of X_j for these (i,j): {1, 254}, {4, 921}, {19, 6504}, {158, 15316}, {2190, 8800}, {13398, 24006}

• X₁₅₅ = crosspoint of X₄ and X₃₅₄₂

• X₁₅₅ = crosssum of X_i and X_j for these (i,j): {3, 15316}, {4, 3147}, {136, 523}, {647, 8754}

• X₁₅₅ = crossdifference of every pair of points on line {924, 2501}

• X₁₅₅ = eigencenter of cevian triangle of X₄

• X₁₅₅ = eigencenter of anticevian triangle of X₃

• X₁₅₅ = X₈₄-of-orthic triangle if ABC is acute

• X₁₅₅ = perspector of orthic triangle and tangential triangle of the MacBeath circumconic, which is also the anticevian triangle of X₃

• X₁₅₅ = perspector of orthic triangle and cross-triangle of ABC and 2nd Hyacinth triangle

• X₁₅₅ = orthic-to-ABC barycentric image of X₃

• X₁₅₅ = intouch-to-ABC functional image of X₄

• X₁₅₅ = barycentric product X_i*X_j for these {i,j}: {3, 6515}, {4, 6503}, {48, 33808}, {63, 920}, {69, 1609}, {343, 8883}, {394, 3542}, {454, 6504}, {3580, 15478}

• X₁₅₅ = barycentric quotient X_i/X_j for these {i,j}: {3, 6504}, {6, 254}, {48, 921}, {216, 8800}, {454, 6515}, {577, 15316}, {920, 92}, {1609, 4}, {3003, 16172}, {3542, 2052}, {6503, 69}, {6515, 264}, {8883, 275}, {15478, 2986}, {18126, 13579}, {32661, 13398}, {33808, 1969}

• X₁₅₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 11411, 12359}, {3, 49, 19357}, {3, 3167, 1147}, {3, 18445, 1181}, {4, 11441, 18451}, {5, 68, 14852}, {5, 12161, 6}, {5, 15068, 17814}, {5, 31831, 1352}, {6, 17814, 5}, {6, 17836, 19458}, {24, 9932, 2931}, {25, 12160, 52}, {25, 12166, 9937}, {26, 156, 154}, {49, 18436, 3}, {51, 21651, 12235}, {52, 10539, 25}, {54, 11459, 7503}, {68, 5654, 5}, {110, 5889, 24}, {143, 13861, 17810}, {154, 17834, 26}, {182, 11793, 7393}, {184, 5562, 3}, {185, 1092, 3}, {185, 3292, 1092}, {381, 12429, 9927}, {389, 9306, 6642}, {394, 1181, 3}, {485, 486, 9722}, {568, 18350, 7506}, {569, 5891, 7395}, {578, 5907, 9818}, {1069, 3157, 1}, {1147, 7689, 12038}, {1147, 12164, 12163}, {1147, 19908, 19357}, {1199, 3090, 5422}, {1351, 1598, 5446}, {1593, 12162, 11472}, {1598, 19588, 12309}, {1614, 11412, 22}, {1993, 6193, 15316}, {1993, 11441, 4}, {3167, 12164, 3}, {3167, 15083, 12163}, {3917, 10984, 3}, {5012, 11444, 7509}, {5020, 11432, 5462}, {5448, 9927, 381}, {5562, 9908, 12163}, {5654, 9936, 68}, {6102, 6644, 9786}, {6193, 22660, 12293}, {6243, 7517, 33586}, {6243, 10540, 7517}, {7395, 11402, 569}, {7488, 9544, 9707}, {7689, 12038, 3}, {9820, 12359, 2}, {10661, 10662, 6}, {10665, 10666, 6}, {11422, 15056, 13434}, {11591, 32046, 7514}, {12161, 15068, 5}, {12162, 13352, 1593}, {12174, 21312, 10575}, {13371, 32140, 1853}, {14516, 22661, 12293}, {15068, 19139, 5654}, {15316, 18451, 12293}, {17814, 17836, 68}, {17814, 19458, 14852}, {18436, 19908, 12163}

Notes

Let (A) be the pedal circle of A wrt the tangential triangle, and define (B), (C) cyclically. The radical center of (A), (B), (C) = X₁₅₅. (Randy Hutson, December 10, 2016)

Let A' be the isogonal conjugate of A with respect to the triangle BCX₄, and define B' and C' cyclically. Let A″B″C″ be the orthic triangle. Then the lines A'A″, B'B″, C'C″ concur in X₁₅₅. (Randy Hutson, 9/23/2011)

Let O_A be the circle centered at the A-vertex of the 2nd anti-extouch triangle and passing through A; define O_B and O_C cyclically. X₁₅₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)