X₁₁₂

Ψ(ORTHOCENTER, SYMMEDIAN POINT)

X₁₁₂ = anticomplement of X₁₂₇

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

X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 32}, {6, 74}

• Related Centers: X₅₂₅ (ISOGONAL CONJUGATE OF X₁₁₂), X₄ (ORTHOCENTER), X₃₂ (3rd POWER POINT), X₆ (SYMMEDIAN POINT), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT)

Trilinears: \(a/(\sin 2B - \sin 2C) : b/(\sin 2C - \sin 2A) : c/(\sin 2A - \sin 2B)\)

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

Tripolars: \(|\cot A \sin(B - C)| : :\)

Search Value (6, 9, 13): \(-0.200462721488\)

▸ Alternate Coordinates2 forms

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

• Trilinears: \(\tan A \csc(B - C) : :\)

▸ Incident Central Lines416 lines

{1, 10705}, {2, 127}, {3, 1297}, {4, 32}, {5, 10749}, {6, 74}, {10, 13280}, {11, 10780}, {12, 13295}, {19, 759}, {20, 10316}, {21, 26703}, {22, 3162}, {23, 14580}, {24, 2079}, {25, 111}, {27, 675}, {28, 105}, {29, 1311}, {30, 2697}, {31, 2249}, {33, 609}, {34, 7031}, {35, 13116}, {36, 13117}, {39, 3520}, {48, 26701}, {50, 477}, {53, 1141}, {54, 217}, {55, 6020}, {56, 3320}, {58, 103}, {69, 2366}, {83, 37125}, {97, 34579}, {99, 648}, {100, 162}, {101, 4249}, {102, 284}, {104, 1108}, {106, 1474}, {107, 1637}, {108, 1415}, {109, 163}, {110, 1625}, {165, 12408}, {172, 6198}, {182, 12207}, {184, 26717}, {186, 187}, {193, 35940}, {216, 11587}, {230, 403}, {246, 19504}, {249, 10425}, {250, 691}, {251, 427}, {264, 2367}, {283, 2365}, {286, 767}, {297, 2857}, {340, 35568}, {371, 19094}, {372, 19093}, {376, 577}, {381, 19163}, {382, 19160}, {384, 1235}, {385, 2868}, {389, 16225}, {393, 571}, {402, 13281}, {420, 8623}, {451, 5277}, {462, 34376}, {463, 34374}, {468, 2770}, {476, 2501}, {493, 13298}, {494, 13299}, {511, 2710}, {512, 2715}, {513, 2722}, {515, 12784}, {516, 2741}, {517, 2747}, {518, 2754}, {519, 2760}, {520, 2764}, {522, 2769}, {523, 935}, {525, 2867}, {574, 14388}, {601, 29056}, {607, 28471}, {631, 34841}, {647, 1304}, {650, 2766}, {651, 13395}, {652, 2762}, {653, 1305}, {662, 1310}, {669, 9091}, {670, 35567}, {681, 36126}, {685, 6037}, {689, 6331}, {692, 36080}, {693, 2859}, {699, 11325}, {729, 1974}, {733, 11380}, {739, 2203}, {741, 1973}, {755, 1843}, {789, 811}, {805, 9218}, {831, 35338}, {835, 1897}, {839, 6335}, {843, 2030}, {898, 5379}, {907, 1634}, {915, 5317}, {917, 8747}, {925, 4226}, {927, 17925}, {930, 35318}, {933, 14586}, {934, 4565}, {953, 3285}, {972, 3194}, {1003, 9308}, {1021, 2765}, {1033, 5897}, {1113, 8105}, {1114, 8106}, {1287, 30716}, {1289, 1632}, {1290, 6591}, {1292, 9523}, {1293, 9527}, {1295, 2193}, {1296, 5467}, {1299, 1609}, {1301, 1624}, {1302, 4240}, {1309, 17926}, {1326, 2700}, {1344, 8426}, {1345, 8427}, {1370, 8879}, {1396, 15728}, {1414, 6183}, {1459, 2727}, {1461, 36079}, {1498, 15324}, {1506, 6143}, {1529, 19158}, {1562, 2777}, {1576, 2492}, {1593, 12145}, {1594, 7745}, {1597, 15433}, {1614, 14585}, {1691, 2211}, {1692, 23700}, {1826, 2372}, {1870, 1914}, {1885, 5305}, {1886, 2688}, {1971, 3331}, {1983, 6011}, {1995, 34108}, {2071, 14961}, {2074, 2752}, {2138, 11413}, {2194, 32726}, {2212, 35106}, {2222, 7115}, {2291, 2299}, {2322, 2370}, {2331, 35192}, {2333, 28482}, {2354, 38453}, {2374, 6353}, {2378, 8739}, {2379, 8740}, {2380, 10642}, {2381, 10641}, {2383, 14576}, {2393, 34107}, {2421, 35575}, {2485, 10423}, {2548, 37119}, {2549, 19220}, {2687, 14571}, {2689, 3064}, {2690, 7649}, {2693, 3284}, {2694, 37960}, {2695, 8755}, {2699, 5006}, {2706, 3289}, {2708, 5060}, {2709, 9181}, {2713, 3049}, {2714, 3063}, {2716, 4282}, {2719, 22383}, {2720, 7252}, {2726, 37168}, {2728, 3737}, {2749, 20752}, {2755, 22356}, {2758, 8756}, {2761, 36054}, {2763, 3292}, {2862, 15149}, {2864, 17924}, {2966, 6528}, {3003, 32710}, {3043, 9696}, {3068, 13923}, {3069, 13992}, {3087, 6128}, {3098, 12503}, {3151, 18686}, {3164, 35952}, {3192, 4262}, {3199, 3518}, {3222, 17941}, {3286, 28838}, {3291, 37777}, {3428, 19159}, {3516, 9605}, {3545, 10314}, {3565, 11634}, {3567, 16224}, {3575, 19157}, {3576, 12265}, {3736, 28844}, {3815, 37118}, {4232, 9084}, {4233, 9061}, {4236, 13397}, {4241, 9057}, {4242, 9070}, {4246, 9058}, {4248, 9083}, {4267, 29206}, {4273, 28159}, {4276, 29045}, {4278, 29042}, {4559, 15439}, {4567, 35574}, {4570, 29241}, {4574, 29163}, {5007, 14865}, {5008, 13596}, {5009, 12032}, {5013, 35477}, {5017, 6403}, {5023, 32534}, {5024, 11410}, {5039, 19124}, {5041, 35478}, {5063, 35485}, {5094, 6032}, {5140, 14659}, {5206, 21844}, {5210, 35472}, {5254, 18560}, {5301, 36420}, {5412, 32420}, {5413, 32422}, {5475, 7577}, {5477, 18331}, {5597, 13229}, {5598, 13231}, {5649, 35911}, {5663, 22146}, {5702, 14482}, {5896, 34570}, {6000, 8779}, {6080, 32320}, {6081, 23090}, {6093, 15471}, {6099, 32698}, {6200, 35828}, {6239, 10881}, {6240, 27376}, {6284, 12955}, {6323, 8541}, {6396, 35829}, {6400, 10880}, {6587, 22239}, {6623, 37689}, {6644, 15355}, {6748, 13597}, {6749, 13338}, {6753, 7468}, {6781, 13619}, {6792, 12828}, {7054, 17512}, {7254, 24016}, {7354, 12945}, {7391, 13854}, {7435, 9107}, {7452, 9056}, {7471, 16167}, {7480, 9060}, {7502, 18472}, {7526, 26216}, {7746, 16868}, {7749, 14940}, {7750, 28724}, {7754, 37199}, {7772, 29322}, {7787, 37337}, {8362, 26224}, {8429, 36176}, {8537, 13330}, {8553, 8746}, {8673, 15388}, {8707, 36797}, {8719, 15576}, {8735, 19628}, {8748, 32706}, {8883, 23233}, {8889, 15437}, {8962, 15218}, {9073, 31905}, {9075, 31909}, {9078, 17520}, {9090, 18344}, {9103, 31902}, {9105, 31903}, {9108, 31900}, {9112, 11612}, {9113, 11613}, {9150, 18020}, {9161, 15463}, {9209, 31510}, {9540, 13918}, {10102, 11580}, {10310, 12340}, {10594, 15745}, {10632, 19781}, {10633, 19780}, {10985, 33842}, {11062, 11063}, {11107, 17916}, {11248, 13118}, {11249, 13119}, {11414, 12413}, {11822, 12478}, {11823, 12479}, {11824, 12805}, {11825, 12806}, {11826, 12925}, {11827, 12935}, {11828, 12996}, {11829, 12997}, {12084, 22120}, {12111, 23128}, {12150, 36794}, {12593, 23701}, {13238, 19127}, {13881, 35488}, {13935, 13985}, {14039, 32000}, {14577, 33643}, {14579, 33631}, {14907, 17907}, {15013, 30737}, {15311, 15341}, {15340, 18400}, {15344, 30733}, {15352, 16813}, {15513, 17506}, {15515, 23040}, {17943, 29329}, {18859, 22121}, {19121, 35924}, {21789, 32642}, {22329, 37855}, {23096, 38463}, {23590, 32646}, {23616, 38233}, {26704, 32653}, {26912, 35949}, {27377, 35920}, {28624, 32739}, {32643, 36067}, {32649, 32687}, {32675, 36078}, {32681, 32715}, {32692, 32734}, {32699, 35182}, {32700, 35183}, {32701, 35184}, {32703, 35185}, {32705, 35186}, {32707, 35187}, {32709, 32729}, {32711, 35189}, {32730, 34397}, {33513, 35178}, {34137, 34146}, {34205, 35569}, {34870, 35476}, {35902, 36201}, {36064, 36131}, {36065, 36104}

▸ Functional Relations41 relations

• X₁₁₂ = reflection of X_i in X_j for these (i,j): (4, 132), (1297, 3)

• X₁₁₂ = isogonal conjugate of X₅₂₅

• X₁₁₂ = isotomic conjugate of X₃₂₆₇

• X₁₁₂ = anticomplement of X₁₂₇

• X₁₁₂ = X_i-Ceva conjugate of X_j for these (i,j): (249, 24), (250, 25)

• X₁₁₂ = cevapoint of X_i and X_j for these (i,j): (32, 512), (427, 523)

• X₁₁₂ = X_i-cross conjugate of X_j for these (i,j): (25, 250), (512, 4), (523, 251)

• X₁₁₂ = crossdifference of every pair of points on line X₁₂₂X₁₂₅

• X₁₁₂ = barycentric product of X₁₁₁₃ and X₁₁₁₄

• X₁₁₂ = isogonal conjugate of isotomic conjugate of trilinear pole of Euler line

• X₁₁₂ = point of intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,X₄,PU(39)}}

• X₁₁₂ = trilinear pole of line X₆X₂₅

• X₁₁₂ = X₆₄₇-cross conjugate of X₆

• X₁₁₂ = pole wrt polar circle of trilinear polar of X₈₅₀ (line X₁₁₅X₁₂₇)

• X₁₁₂ = X₄₈-isoconjugate (polar conjugate) of X₈₅₀

• X₁₁₂ = X₉₂-isoconjugate of X₅₂₀

• X₁₁₂ = X₁₅₇₇-isoconjugate of X₃

• X₁₁₂ = trilinear pole wrt circumsymmedial triangle of line X₆X₆₄₇

• X₁₁₂ = reflection of X₉₃₅ in the Euler line

• X₁₁₂ = reflection of X₂₇₁₅ in the Brocard axis

• X₁₁₂ = reflection of X₂₇₂₂ in line X₁X₃

• X₁₁₂ = inverse-in-polar-circle of X₁₁₅

• X₁₁₂ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₅₆₀

• X₁₁₂ = inverse-in-Moses-radical-circle of X₁₃₀₄

• X₁₁₂ = inverse-in-[circle with diameter X₁₅X₁₆ and center X₁₈₇] of X₈₄₂

• X₁₁₂ = inverse-in-circle-{X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)} of X₂₆₉₈

• X₁₁₂ = inverse-in-Stevanovic-circle of X₂₇₆₆

• X₁₁₂ = X₁₃₉-of-excentral-triangle

• X₁₁₂ = barycentric product of PU(74)

• X₁₁₂ = trilinear product of PU(108)

• X₁₁₂ = eigencenter of circumnormal triangle

• X₁₁₂ = Thomson-isogonal conjugate of X₁₅₀₃

• X₁₁₂ = Lucas-isogonal conjugate of X₁₅₀₃

• X₁₁₂ = perspector of circumcevian triangle of X₄₆₈ and cross-triangle of ABC and circumcevian triangle of X₂₅

• X₁₁₂ = trilinear product of circumcircle intercepts of line X₁X₁₉

• X₁₁₂ = barycentric product X₃*X₁₀₇

• X₁₁₂ = barycentric product X₄*X₁₁₀

• X₁₁₂ = X₆-of-1st-anti-orthosymmedial-triangle

• X₁₁₂ = cevapoint of Jerabek hyperbola intercepts of Lemoine axis

• X₁₁₂ = Ψ(X_i,X_j) for these (i,j): (1, 19), (2, 3), (3, 6), (4, 6), (5, 53), (6, 25), (69, 2), (76, 4), (125, 115)

• X₁₁₂ = Vu circlecevian point of PU(39)

Notes

If the line X₄X₆ is reflected in every side of triangle ABC, then the reflections concur in X₁₁₂. (Randy Hutson, 9/23/2011)

Let P be a point on the van Aubel line other than X₄. Let A' be the reflection of P in BC, and define B' and C' cyclically. The circumcircles of AA'B', BC'A', and CA'B' concur at X₁₁₂. (Randy Hutson, December 26, 2015)

Let Q be a point on the Euler line other than X₂. Let A'B'C' be the cevian triangle of Q. Let A″ be the {B,C}-harmonic conjugate of A' (or equivalently, A″ = BC∩B'C'), and define B″ and C″ cyclically. The circumcircles of AB″C″, BC″A″, CA″B″ concur in X₁₁₂. (Randy Hutson, December 26, 2015)

Let A', B', C' be the intersections of the orthic axis and lines BC, CA, AB, respectively. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₁₂. (Randy Hutson, December 26, 2015)

Let A' be the reflection of X₆ in BC, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₁₂. (Randy Hutson, December 26, 2015)

Let Cm_aCm_bCm_c be the circummedial triangle. Let A″ be the pole, wrt the polar circle, of line Cm_bCm_c, and define B″ and C″ cyclically. The lines Cm_aA″, Cm_bB″, Cm_cC″ concur in X₁₁₂. (Randy Hutson, December 26, 2015)

Let L₁ be the line that is the barycentric product of the Euler line and P(4). Let L₂ be the line that is the barycentric product of the Euler line and U(4). Then X₁₁₂ = L₁∩L₂. (Randy Hutson, July 11, 2019)