X(26) = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE

Trilinears

\(a[b2cos 2B + c2cos 2C - a2cos 2A] : :\)

\((J2 - 3) cos A + 4 cos B cos C : : , where J is as at :ref:`X(1113) <X(1113)>`\)

Barycentrics

\(a2(b2cos 2B + c2cos 2C - a2cos 2A) : :\)

\(a^2 (a^8 - 2 a^6 (b^2 + c^2) + 2 a^2 (b^6 + c^6) - (b^2 - c^2)^2 (b^4 + c^4)) : :\)

Notes

As a point on the Euler line, X(26) has Shinagawa coefficients (E + 4F, -3E - 4F).

If you have The Geometer’s Sketchpad, you can view X(26). If you have GeoGebra, you can view X(26).

Theorems involving X(26), published in 1889 by A. Gob, are discussed in Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960, 259-260.

Let OA be the circle centered at the A-vertex of the circumorthic tangential triangle and passing through A; define OB and OC cyclically. X(26) is the radical center of OA, OB, OC. (Randy Hutson, August 30, 2020)

X(26) lies on these 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}

X(26) is the {X(154),:ref:X(155) <X(155)>}-harmonic conjugate of X(156). For a list of other harmonic conjugates of X(26), click Tables at the top of this page.

X(26) = reflection of X(155) in X(156)

X(26) = isogonal conjugate of X(70)

X(26) = isotomic conjugate of X(20564)

X(26) = tangential isogonal conjugate of X(155)

X(26) = inverse-in-circumcircle of X(2072)

X(26) = crosssum of X(125) and X(924)

X(26) = complement of X(14790)

X(26) = anticomplement of X(13371)

X(26) = intouch-to-ABC functional image of X(3)

X(26) = orthoptic-circle-of-Steiner-inellipse-inverse of complement of X(37978)

X(26) = orthoptic-circle-of-Steiner-circumellipse-inverse of anticomplement of X(37978)