X₂₆

CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Theorems involving X₂₆, published in 1889 by A. Gob, are discussed in

Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

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

X₂₆ (CIRCUMCENTER OF THE TANGENTIAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Tangential Triangle △Tan_aTan_bTan_c — vertices Tan_a, Tan_b, Tan_c (Bounded by tangents to the circumcircle at A, B, C)

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Circle: circumcircle_oftangential — Centered at X₂₆

• Concurrence Lines: Tan_a–X₂₆, Tan_b–X₂₆, Tan_c–X₂₆ concurring at X₂₆

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₂ (CENTROID)

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

Barycentrics: \(a^{2}(b^{2}\cos 2B + c^{2}\cos 2C - a^{2}\cos 2A) : :\)

Search Value (6, 9, 13): \(-2.116776034478\)

▸ Alternate Coordinates2 forms

• Trilinears: \((J^{2} - 3) \cos A + 4 \cos B \cos C : :\), where J is as at X₁₁₁₃

• Barycentrics: \(a^{2} (a^{8} - 2 a^{6} (b^{2} + c^{2}) + 2 a^{2} (b^{6} + c^{6}) - (b^{2} - c^{2})^{2} (b^{4} + c^{4})) : :\)

▸ Linear Combinations2 combinations

• X₂₆ = \(6\) X₂ + \((J^{2} - 7)\) X₃

• X₂₆ = \((J^{2} - 3)\) X₃ + \(2\) X₄

▸ Incident Central Lines318 lines

{1, 9625}, {2, 3}, {6, 143}, {10, 9712}, {39, 9608}, {40, 9590}, {49, 1993}, {51, 569}, {52, 184}, {54, 3060}, {55, 4354}, {56, 4351}, {64, 32138}, {68, 161}, {74, 12279}, {98, 1286}, {110, 7731}, {113, 22109}, {154, 155}, {157, 2934}, {159, 3564}, {182, 5462}, {195, 9704}, {197, 32141}, {206, 511}, {221, 32143}, {232, 10316}, {343, 12134}, {355, 8185}, {394, 6101}, {495, 10037}, {496, 10046}, {512, 39537}, {524, 9925}, {542, 15581}, {568, 6800}, {577, 14576}, {578, 5446}, {912, 40660}, {952, 9798}, {970, 9570}, {1092, 10625}, {1112, 12228}, {1151, 9682}, {1177, 14984}, {1181, 6102}, {1199, 11003}, {1204, 10575}, {1209, 32332}, {1216, 9306}, {1350, 10627}, {1351, 14449}, {1352, 20987}, {1353, 19459}, {1478, 9658}, {1479, 9672}, {1483, 8192}, {1493, 19468}, {1495, 5562}, {1498, 2931}, {1503, 12359}, {1511, 35602}, {1601, 25043}, {1602, 35220}, {1603, 35221}, {1605, 1607}, {1606, 1608}, {1609, 42459}, {1614, 5889}, {1829, 24301}, {1843, 19131}, {1853, 13561}, {1971, 23128}, {1974, 9967}, {2079, 5023}, {2165, 8553}, {2192, 32168}, {2351, 31381}, {2393, 8548}, {2777, 12893}, {2781, 15132}, {2782, 39828}, {2794, 39825}, {2916, 5085}, {2929, 35237}, {2935, 34584}, {3070, 35776}, {3071, 35777}, {3098, 5447}, {3197, 32158}, {3205, 36979}, {3206, 36981}, {3220, 24467}, {3357, 14915}, {3425, 10547}, {3527, 13451}, {3532, 34802}, {3556, 14988}, {3567, 5012}, {3580, 25738}, {3581, 11456}, {3796, 5946}, {5092, 11695}, {5285, 26921}, {5347, 36754}, {5412, 10898}, {5413, 10897}, {5422, 13353}, {5448, 23358}, {5449, 18381}, {5594, 5874}, {5595, 5875}, {5621, 20379}, {5640, 38848}, {5690, 8193}, {5844, 12410}, {5876, 18451}, {5890, 37490}, {5891, 44082}, {5892, 37515}, {5901, 11365}, {5907, 32237}, {5944, 10263}, {6000, 7689}, {6030, 15045}, {6146, 32269}, {6237, 10536}, {6238, 10535}, {6247, 44158}, {6403, 19121}, {6407, 9694}, {6515, 32358}, {6759, 9932}, {6776, 18951}, {7293, 37612}, {7352, 26888}, {7691, 11459}, {7712, 15032}, {7742, 14667}, {7767, 15574}, {8190, 32146}, {8191, 32147}, {8194, 32177}, {8195, 32178}, {8276, 8981}, {8277, 13966}, {8538, 44102}, {8550, 35707}, {8718, 15072}, {8743, 10317}, {8746, 36418}, {9143, 25714}, {9730, 10984}, {9781, 13434}, {9786, 13630}, {9820, 10192}, {9861, 9918}, {9911, 28174}, {9915, 22657}, {9916, 22656}, {9917, 22655}, {9920, 12310}, {9927, 18400}, {9938, 13289}, {10095, 17810}, {10113, 19457}, {10182, 29317}, {10264, 13171}, {10312, 22240}, {10313, 22120}, {10533, 10665}, {10534, 10666}, {10540, 11441}, {10601, 15026}, {10605, 13491}, {10606, 32210}, {10610, 10982}, {10628, 40276}, {10632, 11421}, {10633, 11420}, {10634, 10642}, {10635, 10641}, {10661, 30402}, {10662, 30403}, {10663, 10682}, {10664, 10681}, {10733, 40242}, {10790, 32134}, {10828, 32151}, {10829, 10943}, {10830, 10942}, {10833, 15171}, {10834, 32213}, {10835, 32214}, {10880, 11418}, {10881, 11417}, {11202, 12038}, {11206, 11411}, {11248, 20872}, {11399, 37729}, {11402, 37493}, {11430, 13598}, {11432, 16881}, {11438, 40647}, {11440, 12290}, {11444, 43598}, {11449, 43574}, {11455, 15062}, {11464, 15107}, {11468, 13445}, {11472, 15811}, {11477, 13421}, {11482, 43697}, {11499, 20989}, {11591, 17814}, {11645, 14864}, {11671, 34418}, {12006, 37514}, {12111, 14157}, {12160, 26864}, {12162, 26883}, {12164, 14530}, {12220, 19128}, {12236, 13198}, {12280, 12380}, {12289, 41482}, {12293, 17845}, {12295, 32607}, {12307, 41726}, {12370, 19467}, {12891, 13288}, {12892, 13287}, {13142, 43595}, {13292, 31804}, {13336, 22352}, {13352, 13367}, {13391, 17821}, {13419, 21243}, {13558, 15653}, {13562, 37485}, {13567, 18952}, {13889, 13925}, {13943, 13993}, {14128, 33533}, {14531, 43844}, {14641, 43604}, {14657, 33962}, {14693, 32762}, {15035, 25487}, {15043, 15080}, {15067, 35259}, {15069, 19596}, {15085, 17838}, {15172, 16541}, {15454, 16104}, {15462, 40949}, {15478, 34428}, {15912, 40947}, {15959, 25150}, {16010, 35218}, {16165, 25711}, {16252, 22660}, {16391, 23181}, {17809, 32136}, {17811, 32142}, {17813, 32155}, {17819, 32169}, {17820, 32170}, {17825, 32205}, {17826, 32207}, {17827, 32208}, {18350, 23039}, {18376, 32393}, {18379, 18405}, {18874, 31860}, {18954, 18990}, {19005, 19116}, {19006, 19117}, {19129, 39588}, {19132, 19155}, {19165, 20993}, {19180, 19211}, {19189, 19210}, {19194, 26887}, {19908, 32048}, {20191, 23329}, {20299, 29012}, {20424, 32333}, {20477, 44138}, {20771, 41673}, {20791, 43597}, {21651, 34750}, {21849, 37505}, {22115, 37484}, {22533, 22550}, {22654, 32153}, {23698, 39854}, {23709, 34292}, {26446, 37557}, {29181, 35228}, {32613, 39582}, {32620, 33537}, {32829, 44180}, {34116, 44078}, {34118, 34177}, {34380, 37491}, {34397, 35603}, {34417, 37513}, {34514, 34826}, {35219, 39879}, {35719, 41244}, {39805, 39835}, {39806, 39834}, {39823, 39853}, {39824, 39852}, {39829, 39859}, {39830, 39858}, {43575, 43829}

▸ Functional Relations11 relations

• X₂₆ = reflection of X₁₅₅ in X₁₅₆

• X₂₆ = isogonal conjugate of X₇₀

• X₂₆ = isotomic conjugate of X₂₀₅₆₄

• X₂₆ = tangential isogonal conjugate of X₁₅₅

• X₂₆ = inverse-in-circumcircle of X₂₀₇₂

• X₂₆ = crosssum of X₁₂₅ and X₉₂₄

• X₂₆ = complement of X₁₄₇₉₀

• X₂₆ = anticomplement of X₁₃₃₇₁

• X₂₆ = intouch-to-ABC functional image of X₃

• X₂₆ = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X₃₇₉₇₈

• X₂₆ = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X₃₇₉₇₈

Notes

As a point on the Euler line, X₂₆ has Shinagawa coefficients \((E + 4F, -3E - 4F)\).

Let O_A be the circle centered at the A-vertex of the circumorthic tangential triangle and passing through A; define O_B and O_C cyclically. X₂₆ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₂₆ is the {X₁₅₄,X₁₅₅}-harmonic conjugate of X₁₅₆.