X25

HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES

Constructed as indicated by the name; also X25 is the pole of the orthic axis (the line having trilinear coefficients cos A : cos B : cos C) with respect to the circumcircle.
< 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) { 2, 3 }: Euler Line { 19, 33 }: Central Line L(19, 33) { 6, 51 }: Central Line L(6, 51) { 41, 42 }: Central Line L(41, 42) { 34, 56 }: Central Line L(34, 56) { 98, 107 }: Central Line L(98, 107) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_19 = CLAWSON POINT (0.6246, 0.4244) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_41 = -CEVA CONJUGATE OF X(31) (0.5823, 0.2244) X_42 = CROSSPOINT OF INCENTER AND SYMMEDIAN POINT (0.5880, 0.2495) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_98 = TARRY POINT (0.8891, -0.2281) X_107 = Ψ(SYMMEDIAN POINT, ORTHOCENTER) (0.5733, 0.6071) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) A B C X69 X2 X3 X19 X33 X6 X51 X41 X42 X34 X56 X98 X107 X25 (a, b, c) = (5, 6, 7) | AB = 1

X25 (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sin A \tan A : :\)
Barycentrics: \(\tan A - \tan \omega : :\)
Search Value (6, 9, 13): \(-0.546404736966\)
▸ Alternate Coordinates3 forms

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

  • Trilinears: \(\cos A - \sec A : :\)

  • Barycentrics: \(\sec A \sin(A - \omega ) : :\)

▸ Linear Combinations1 combination

  • X25 = \(6 R^{2}\) X2 - \(S_{\omega}\) X3

▸ Incident Central Lines838 lines

{1, 1036}, {2, 3}, {6, 51}, {7, 7717}, {8, 7718}, {9, 5285}, {10, 5090}, {11, 10829}, {12, 10830}, {13, 9916}, {14, 9915}, {17, 22482}, {18, 22481}, {19, 33}, {31, 608}, {32, 1184}, {34, 56}, {35, 1900}, {36, 1878}, {39, 39951}, {40, 1902}, {41, 42}, {43, 37576}, {48, 14547}, {49, 36749}, {52, 155}, {53, 157}, {54, 3527}, {57, 1473}, {58, 967}, {63, 24320}, {64, 1192}, {65, 1452}, {66, 34207}, {67, 32239}, {68, 9908}, {69, 8263}, {72, 37547}, {74, 3426}, {76, 1241}, {79, 16114}, {80, 9912}, {81, 37492}, {83, 9918}, {84, 9910}, {92, 242}, {97, 14489}, {98, 107}, {99, 2374}, {100, 1862}, {101, 3190}, {104, 3420}, {105, 108}, {106, 9088}, {110, 1112}, {111, 112}, {113, 2931}, {114, 135}, {115, 3455}, {125, 1853}, {132, 136}, {133, 14703}, {137, 15959}, {141, 37485}, {143, 156}, {146, 34796}, {160, 3815}, {162, 37128}, {171, 7295}, {181, 2175}, {182, 3066}, {183, 264}, {185, 1498}, {187, 5140}, {190, 24814}, {193, 6339}, {195, 2904}, {200, 4006}, {209, 2911}, {210, 12329}, {212, 2183}, {216, 10314}, {219, 26885}, {220, 3690}, {221, 1425}, {222, 20122}, {225, 1842}, {226, 1892}, {236, 8132}, {238, 5329}, {244, 36570}, {250, 5968}, {251, 5359}, {262, 275}, {263, 2211}, {265, 12140}, {269, 40223}, {273, 1447}, {281, 7102}, {286, 1218}, {305, 683}, {317, 325}, {318, 7081}, {324, 42354}, {339, 1289}, {340, 7788}, {343, 1352}, {351, 14998}, {353, 11226}, {354, 22769}, {371, 493}, {372, 494}, {373, 5085}, {385, 2998}, {387, 38879}, {389, 1181}, {393, 1033}, {394, 511}, {399, 1986}, {459, 3424}, {476, 38552}, {485, 9922}, {486, 9921}, {487, 12169}, {488, 12170}, {497, 16541}, {512, 2433}, {516, 21062}, {518, 41611}, {524, 41585}, {542, 12828}, {543, 2936}, {568, 10540}, {571, 39109}, {573, 2328}, {574, 15433}, {576, 21849}, {577, 26953}, {578, 10110}, {581, 2360}, {588, 3311}, {589, 3312}, {597, 35707}, {599, 34992}, {604, 2208}, {610, 10382}, {647, 34212}, {648, 3228}, {667, 18344}, {669, 878}, {671, 9876}, {675, 2973}, {685, 40820}, {691, 40119}, {692, 913}, {694, 1613}, {800, 15259}, {842, 1304}, {847, 1179}, {884, 6591}, {895, 41616}, {908, 27388}, {915, 9058}, {917, 9057}, {925, 2974}, {933, 5966}, {935, 2770}, {940, 18165}, {941, 1172}, {954, 21319}, {958, 1891}, {973, 32341}, {974, 9934}, {982, 36572}, {993, 5155}, {999, 1870}, {1001, 1848}, {1007, 9723}, {1030, 39982}, {1062, 9645}, {1068, 26228}, {1073, 1297}, {1084, 16098}, {1092, 37498}, {1093, 8794}, {1096, 1402}, {1108, 40970}, {1119, 3598}, {1141, 15960}, {1147, 5446}, {1151, 6291}, {1152, 6406}, {1154, 15068}, {1162, 8904}, {1163, 8903}, {1164, 8974}, {1165, 13950}, {1169, 2189}, {1173, 43908}, {1177, 32246}, {1180, 3108}, {1216, 37486}, {1235, 1239}, {1249, 5304}, {1261, 7046}, {1291, 23096}, {1298, 22551}, {1299, 34333}, {1300, 1302}, {1309, 2726}, {1311, 21666}, {1324, 1785}, {1327, 13668}, {1328, 13788}, {1329, 9712}, {1350, 3917}, {1376, 1861}, {1383, 5354}, {1396, 42290}, {1397, 3271}, {1403, 8852}, {1407, 3937}, {1436, 7008}, {1437, 36742}, {1441, 26260}, {1466, 22344}, {1470, 1877}, {1482, 41722}, {1501, 1976}, {1503, 1619}, {1506, 9700}, {1511, 15472}, {1560, 2079}, {1565, 39732}, {1576, 2493}, {1604, 1863}, {1605, 6108}, {1606, 6109}, {1610, 3486}, {1611, 1968}, {1614, 3567}, {1620, 8567}, {1622, 38870}, {1626, 20470}, {1631, 1826}, {1633, 3474}, {1634, 9766}, {1637, 42659}, {1659, 30386}, {1698, 9591}, {1699, 9590}, {1716, 7093}, {1726, 1736}, {1730, 1754}, {1753, 10310}, {1790, 37474}, {1825, 23844}, {1830, 23845}, {1831, 23846}, {1838, 7742}, {1839, 8053}, {1840, 23851}, {1841, 2178}, {1857, 42069}, {1864, 2182}, {1866, 26437}, {1867, 39585}, {1869, 37601}, {1871, 10267}, {1872, 11248}, {1887, 11509}, {1897, 8851}, {1946, 40134}, {1989, 6103}, {1990, 5306}, {1994, 5093}, {2004, 36759}, {2005, 36760}, {2051, 9570}, {2053, 4426}, {2056, 13330}, {2076, 21001}, {2155, 2357}, {2181, 6187}, {2192, 3270}, {2198, 14974}, {2217, 3435}, {2262, 18621}, {2270, 7070}, {2332, 4258}, {2350, 5021}, {2353, 3767}, {2489, 6041}, {2502, 11173}, {2548, 23208}, {2697, 22239}, {2752, 2766}, {2781, 15106}, {2793, 34519}, {2854, 6096}, {2881, 42665}, {2883, 13568}, {2886, 9713}, {2914, 5898}, {2916, 31521}, {2917, 3574}, {2929, 5895}, {2930, 5095}, {2932, 5151}, {2934, 2963}, {2935, 13202}, {2968, 26703}, {2979, 10546}, {2981, 10632}, {3052, 8750}, {3064, 23865}, {3068, 13884}, {3069, 13937}, {3087, 7736}, {3098, 3819}, {3168, 9755}, {3197, 3611}, {3202, 27375}, {3259, 10016}, {3286, 39984}, {3292, 11470}, {3295, 3920}, {3305, 5314}, {3306, 7293}, {3357, 13474}, {3407, 37892}, {3425, 6530}, {3431, 3531}, {3456, 7755}, {3506, 35431}, {3511, 23173}, {3519, 41598}, {3532, 22334}, {3564, 6515}, {3572, 43925}, {3580, 11442}, {3581, 18435}, {3589, 3867}, {3679, 37546}, {3695, 5687}, {3705, 5081}, {3772, 23847}, {3794, 26625}, {3818, 21243}, {3868, 42461}, {3924, 8615}, {3933, 40123}, {4057, 7649}, {4108, 14618}, {4265, 37674}, {4383, 5347}, {4650, 24436}, {5012, 5050}, {5024, 37808}, {5092, 6688}, {5096, 37679}, {5120, 33854}, {5121, 40293}, {5146, 29681}, {5152, 32527}, {5174, 29641}, {5201, 8667}, {5203, 5866}, {5204, 5370}, {5210, 8585}, {5217, 7302}, {5248, 39579}, {5254, 9608}, {5292, 5358}, {5305, 41361}, {5307, 16678}, {5311, 17442}, {5324, 37642}, {5342, 16823}, {5364, 20678}, {5406, 9739}, {5407, 9738}, {5408, 9733}, {5409, 9732}, {5418, 9683}, {5462, 36752}, {5476, 32267}, {5480, 10192}, {5504, 20771}, {5512, 14657}, {5521, 14667}, {5523, 5938}, {5544, 6030}, {5562, 17814}, {5584, 11471}, {5587, 9625}, {5593, 18130}, {5596, 26926}, {5597, 8190}, {5598, 8191}, {5622, 12099}, {5644, 7712}, {5647, 42445}, {5650, 31884}, {5707, 18180}, {5889, 11441}, {5890, 11456}, {5891, 37478}, {5913, 21397}, {5926, 39533}, {5965, 41599}, {5986, 12188}, {5989, 6331}, {6000, 10605}, {6054, 20774}, {6088, 10103}, {6094, 33900}, {6102, 32139}, {6114, 31688}, {6115, 31687}, {6145, 32332}, {6146, 9833}, {6151, 10633}, {6193, 12309}, {6197, 10306}, {6200, 41438}, {6217, 19352}, {6218, 19351}, {6239, 12313}, {6241, 12315}, {6242, 12316}, {6243, 18350}, {6247, 16621}, {6289, 12973}, {6290, 12972}, {6344, 31676}, {6391, 12272}, {6396, 41437}, {6400, 12314}, {6423, 8577}, {6424, 8576}, {6561, 9682}, {6564, 8280}, {6565, 8281}, {6696, 16656}, {6749, 9300}, {6750, 15512}, {6751, 17849}, {6752, 41373}, {6753, 8651}, {6767, 29815}, {6776, 11206}, {7017, 17987}, {7028, 8131}, {7079, 40175}, {7160, 12139}, {7179, 7282}, {7283, 19799}, {7373, 17024}, {7607, 39284}, {7612, 8796}, {7664, 34517}, {7665, 8878}, {7687, 13289}, {7691, 15056}, {7722, 12308}, {7745, 15270}, {7746, 27371}, {7762, 19597}, {7766, 38262}, {7773, 37804}, {7774, 20794}, {7779, 22152}, {7784, 21248}, {7792, 17907}, {7828, 33802}, {7952, 39600}, {8024, 22241}, {8071, 15654}, {8105, 42668}, {8106, 42667}, {8157, 10214}, {8227, 9626}, {8266, 15271}, {8267, 22253}, {8278, 32577}, {8537, 11422}, {8550, 15581}, {8584, 15471}, {8588, 33880}, {8735, 23402}, {8745, 8882}, {8746, 14577}, {8749, 20975}, {8756, 15621}, {8840, 17984}, {8887, 31381}, {8901, 19174}, {8911, 26868}, {8939, 19404}, {8943, 19405}, {9056, 32706}, {9059, 40101}, {9060, 32710}, {9061, 26706}, {9070, 39439}, {9083, 32704}, {9084, 30247}, {9135, 11631}, {9209, 39201}, {9475, 38867}, {9659, 10895}, {9672, 10896}, {9694, 43512}, {9695, 43509}, {9704, 14627}, {9708, 29667}, {9709, 29679}, {9748, 38918}, {9756, 42400}, {9792, 19170}, {9822, 19126}, {9927, 19908}, {9932, 22660}, {9935, 11577}, {9993, 14165}, {10095, 32046}, {10098, 10102}, {10111, 12419}, {10263, 16266}, {10266, 12146}, {10272, 11566}, {10274, 11808}, {10278, 41357}, {10313, 15355}, {10317, 36414}, {10478, 17188}, {10519, 33522}, {10535, 11436}, {10536, 11435}, {10545, 11451}, {10571, 41401}, {10606, 21663}, {10620, 12292}, {10643, 11516}, {10644, 11515}, {10961, 11514}, {10963, 11513}, {10974, 16471}, {10984, 37514}, {11003, 34545}, {11064, 31670}, {11174, 36794}, {11175, 20965}, {11179, 20192}, {11188, 41614}, {11202, 11430}, {11412, 43598}, {11424, 11425}, {11427, 14853}, {11439, 11440}, {11444, 43614}, {11457, 16659}, {11459, 33523}, {11464, 15033}, {11472, 16194}, {11475, 11480}, {11476, 11481}, {11487, 16543}, {11574, 19137}, {11580, 40103}, {11695, 37515}, {11743, 32391}, {11745, 12233}, {11746, 13198}, {11817, 15047}, {12007, 15580}, {12162, 12163}, {12220, 26206}, {12228, 20773}, {12235, 19458}, {12236, 19456}, {12237, 19461}, {12238, 19462}, {12239, 19463}, {12240, 19464}, {12241, 15873}, {12242, 19468}, {12279, 43601}, {12290, 13093}, {12293, 12301}, {12295, 12302}, {12296, 12303}, {12297, 12304}, {12298, 12305}, {12299, 12306}, {12300, 12307}, {12311, 12509}, {12312, 12510}, {12324, 18913}, {12335, 40953}, {12420, 12421}, {12429, 14516}, {13007, 13051}, {13008, 13052}, {13013, 19465}, {13014, 19466}, {13019, 13021}, {13020, 13022}, {13023, 13035}, {13024, 13036}, {13148, 14094}, {13233, 36523}, {13321, 15087}, {13336, 15805}, {13346, 13598}, {13390, 30385}, {13394, 14561}, {13403, 34785}, {13417, 17847}, {13419, 18381}, {13450, 34449}, {13507, 13597}, {13754, 18451}, {13851, 18405}, {13858, 36330}, {13859, 35752}, {14216, 16655}, {14264, 35372}, {14458, 16080}, {14490, 43713}, {14492, 43530}, {14529, 42450}, {14535, 32581}, {14683, 18947}, {14713, 14715}, {14810, 16187}, {14845, 37513}, {14852, 18474}, {14855, 35237}, {15053, 15072}, {15069, 41586}, {15121, 41603}, {15126, 15127}, {15135, 34117}, {15139, 37473}, {15141, 38851}, {15300, 33850}, {15302, 38862}, {15321, 34436}, {15462, 16165}, {15463, 32609}, {15475, 15551}, {15589, 32000}, {15591, 40321}, {15651, 40052}, {15655, 20481}, {15668, 17171}, {15740, 43690}, {15741, 32605}, {16010, 32250}, {16178, 16188}, {16221, 42426}, {16231, 39225}, {16263, 22455}, {16277, 43678}, {16317, 36878}, {16583, 18616}, {16776, 19127}, {16778, 39954}, {16817, 19798}, {16835, 43719}, {16974, 21010}, {17054, 24163}, {17808, 40124}, {17824, 32352}, {17830, 35711}, {17835, 21650}, {17836, 21651}, {17837, 21652}, {17838, 21649}, {17839, 21653}, {17840, 21655}, {17841, 21657}, {17842, 21654}, {17843, 21656}, {17844, 21658}, {17845, 21659}, {17846, 21660}, {17924, 26249}, {17983, 18818}, {18020, 31632}, {18390, 18396}, {18475, 37506}, {18613, 23710}, {18615, 22363}, {18619, 41015}, {18651, 24701}, {18755, 39967}, {18906, 37894}, {18907, 41370}, {18909, 34781}, {18912, 34224}, {18914, 18916}, {18928, 25406}, {18950, 39874}, {18997, 19039}, {18998, 19040}, {19140, 40291}, {19149, 19161}, {19169, 19172}, {19180, 21638}, {19349, 19366}, {19358, 19410}, {19359, 19411}, {19418, 19424}, {19419, 19425}, {19430, 21642}, {19431, 21643}, {19460, 22530}, {19583, 40324}, {19724, 19756}, {19725, 19763}, {20032, 20034}, {20266, 26933}, {20271, 21771}, {20423, 35266}, {20468, 42071}, {20791, 43584}, {21148, 40934}, {21461, 34394}, {21462, 34395}, {21661, 22552}, {21850, 37645}, {21851, 34779}, {22080, 37499}, {22109, 36518}, {22240, 23635}, {22331, 36616}, {22466, 22483}, {22538, 22549}, {22662, 22953}, {23039, 37494}, {23180, 36849}, {23224, 42772}, {23291, 32064}, {23359, 41011}, {23361, 26357}, {23675, 28037}, {23858, 42070}, {24682, 25343}, {24686, 25344}, {24855, 43618}, {26227, 41013}, {26235, 44142}, {26262, 38462}, {26266, 44143}, {26269, 41375}, {26275, 39200}, {26877, 26928}, {26878, 26938}, {26886, 26894}, {26907, 26909}, {26918, 26936}, {27365, 41615}, {27370, 40643}, {28476, 32691}, {28782, 28783}, {30249, 34168}, {30687, 31394}, {30737, 44131}, {31382, 35709}, {32001, 37668}, {32078, 40674}, {32137, 32138}, {32145, 32166}, {32234, 32254}, {32260, 32276}, {32264, 32285}, {32340, 32345}, {32359, 32377}, {32445, 40951}, {32654, 39072}, {32674, 34068}, {32676, 34079}, {32929, 42707}, {33801, 35222}, {33863, 39966}, {33873, 35458}, {34096, 41278}, {34382, 41619}, {34448, 41221}, {34482, 39955}, {34783, 37490}, {34803, 44180}, {35012, 36067}, {35219, 36851}, {35278, 39656}, {36743, 39798}, {36901, 44176}, {36983, 43616}, {37483, 43586}, {37502, 39971}, {37503, 39974}, {37644, 39899}, {37665, 40065}, {37667, 43981}, {37671, 44134}, {37778, 40102}, {38292, 41894}, {38920, 41414}, {38956, 40082}, {39111, 39112}, {39417, 40358}, {39530, 41244}, {39644, 39645}, {39646, 40814}, {39806, 39810}, {39809, 39812}, {39817, 39820}, {39835, 39839}, {39838, 39841}, {39846, 39849}, {40116, 43079}, {40169, 40184}, {40182, 40195}, {40185, 42484}, {40187, 40189}, {40190, 40219}, {40220, 40226}, {40285, 41725}, {40316, 40317}, {40454, 43742}, {41410, 41445}, {41411, 41444}, {42394, 44144}, {43460, 43462}, {43725, 43726}

▸ Functional Relations70 relations

Notes

As a point on the Euler line, X25 has Shinagawa coefficients \((F, -E - F)\).

Let A' be the center of the conic through the contact points of the incircle and the A-excircle with the sidelines of ABC. Define B' and C' cyclically. Let A″ be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B″ and C″ cyclically. Let A* be the midpoint of A' and A″, and define B* and C* cyclically. The triangle A*B*C* is perspective to ABC at X25. See also X6, X218, X222, X940, X1743. (Randy Hutson, July 23, 2015)

Let A' be the radical center of the nine-point circle and the B- and C-power circles. efine B' and C' cyclically. The triangle A'B'C' is homothetic with the orthic triangle, and the center of homothety is X25. Also X25 is the point of intersection of these two lines: isotomic conjugate of polar conjugate of van Aubel line (i.e., line X2X3), and polar conjugate of isotomic conjugate of van Aubel line (i.e., line X25X393). Also, X25 is the trilinear pole of line X512X1692, this line being the isogonal conjugate of the isotomic conjugate of the orthic axis; the line X512X1692 is also the polar of X76 wrt polar circle, and the line is also the radical axis of circumcircle and 2nd Lemoine circle. (Randy Hutson, September 5, 2015)

Let HaHbHc be the orthic triangle. Let A″ be the barycentric product of the (real or imaginary) circumcircle intercepts of line HbHc. Define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X25. (Randy Hutson, October 27, 2015)

The 2nd Ehrmann triangle, defined in the preamble to X8537, can be generalized as follows. Let P be a point in the plane of ABC and not on BC∪CA∪AB. Let Ab the the point of intersection of the circle {P,B,C}} and the line AB, and define Bc and Ca cyclically. Define Ac symmetrically, and define Ba and Cb cyclically. Let A' = BcBa∩CaCb, and define B' and C' cyclically. Triangle A'B'C', here introduced as the P-Ehrmann triangle, is homothetic to the orthic triangle. The X1-Ehrmann triangle is the intangents triangle, and the X6-Ehrmann triangle is the 2nd Ehrmann triangle. If P lies on the circumcircle, the P-Ehrmann triangle is the tangential triangle. If P is on the Brocard 2nd cubic K018 or the circumcircle, then the P-Ehrmann triangle is perspective to ABC. The homothetic center of the orthic triangle and the X4-Ehrmann triangle is X25. (Randy Hutson, February 10, 2016)

Let H(X) denote hodpoint of a point X, as defined in the preamble just before X40139. Then H(X2) = H(X8115) = H(X8116) = X25. (Radosław Żak, October 29, 2020)

Let Ga = reflection of X6 in line BC, and define Gb and Gc cyclically. Let T = tangential triangle and H = orthic triangle. Then each pair of the triangles GaGbGc, T, and H are perspective, and the perspector in all three cases is X25. (Dasari Naga Vijay Krishna, March 14, 2021)

X25 is the {X5,X26}-harmonic conjugate of X3.