X₁₀₇

Ψ(SYMMEDIAN POINT, ORTHOCENTER)

X₁₀₇ = anticomplement of X₁₂₂

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

X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 74}, {25, 98}

• Related Centers: X₄ (ORTHOCENTER), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₉₈ (TARRY POINT)

Trilinears: \(1/[\cos A (\sin 2B - \sin 2C)] : :\)

Barycentrics: \(1/[(b^{2} - c^{2})(b^{2} + c^{2} - a^{2})^{2}] : 1/[(c^{2} - a^{2})(c^{2} + a^{2} - b^{2})^{2}] : 1/[(a^{2} - b^{2})(a^{2} + b^{2} - c^{2})^{2}]\)

Tripolars: \(|(\cos A)(\tan B - \tan C)| : :\)

Search Value (6, 9, 13): \(0.245357987936\)

▸ Alternate Coordinates2 forms

• Trilinears: \((\sec A)/(\tan B - \tan C) : :\)

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

▸ Incident Central Lines227 lines

{1, 10701}, {2, 122}, {3, 1294}, {4, 74}, {5, 10745}, {6, 10762}, {11, 10775}, {19, 2249}, {20, 3184}, {21, 1295}, {22, 15466}, {23, 2697}, {24, 1093}, {25, 98}, {27, 103}, {28, 104}, {29, 102}, {30, 2693}, {51, 275}, {55, 7158}, {56, 3324}, {76, 2366}, {83, 27373}, {92, 26702}, {99, 2797}, {100, 823}, {101, 1897}, {105, 2833}, {106, 2839}, {108, 2845}, {109, 162}, {110, 648}, {111, 393}, {112, 1637}, {154, 15576}, {158, 759}, {184, 3168}, {186, 477}, {243, 14192}, {250, 687}, {264, 1995}, {284, 8764}, {297, 2710}, {324, 13595}, {333, 2365}, {373, 37124}, {376, 36876}, {382, 23241}, {403, 11657}, {415, 2708}, {419, 2698}, {422, 2699}, {423, 2700}, {425, 2707}, {427, 29011}, {428, 29316}, {450, 511}, {458, 3066}, {459, 32064}, {460, 23700}, {468, 842}, {470, 7684}, {471, 7685}, {472, 5479}, {473, 5478}, {476, 7480}, {512, 2713}, {513, 2719}, {514, 2727}, {515, 2732}, {516, 2738}, {517, 2744}, {518, 2749}, {519, 2755}, {520, 6080}, {521, 2761}, {522, 2762}, {523, 1304}, {524, 2763}, {525, 2764}, {631, 34842}, {685, 2501}, {691, 7473}, {699, 36417}, {729, 2207}, {739, 5317}, {741, 1096}, {755, 27376}, {811, 1310}, {841, 10295}, {850, 2867}, {859, 2734}, {877, 4563}, {915, 30733}, {925, 15329}, {930, 38342}, {933, 1576}, {934, 13149}, {935, 14618}, {953, 37168}, {972, 4183}, {1075, 6759}, {1105, 22467}, {1112, 9161}, {1141, 3518}, {1172, 32726}, {1249, 6793}, {1290, 37966}, {1292, 4238}, {1293, 9524}, {1296, 4235}, {1299, 3542}, {1305, 4243}, {1325, 2694}, {1559, 15311}, {1624, 1632}, {1625, 6570}, {1651, 34601}, {1783, 36080}, {1857, 28471}, {1981, 36516}, {1990, 15448}, {2073, 2688}, {2074, 2687}, {2075, 2695}, {2291, 8748}, {2367, 18027}, {2370, 7419}, {2374, 21447}, {2382, 34856}, {2404, 9064}, {2489, 9091}, {2691, 7476}, {2696, 7482}, {2714, 18344}, {2716, 17515}, {2722, 17924}, {2724, 37908}, {2752, 37963}, {2770, 37778}, {2868, 17984}, {3079, 11206}, {3090, 36520}, {3124, 6531}, {3146, 5896}, {3183, 6225}, {3563, 6353}, {3565, 4226}, {3616, 11732}, {3658, 13397}, {3952, 29163}, {4228, 26703}, {4242, 6011}, {4244, 26706}, {5064, 29322}, {5094, 14388}, {5379, 6099}, {5502, 33640}, {5640, 36794}, {5966, 14569}, {5994, 36309}, {5995, 36306}, {6015, 6059}, {6081, 7253}, {6587, 23590}, {6618, 11433}, {6761, 18400}, {6995, 29180}, {7009, 15168}, {7192, 24016}, {7337, 35108}, {7426, 37765}, {7488, 18284}, {7493, 17907}, {7722, 16169}, {7728, 11251}, {8057, 15384}, {8749, 13479}, {8750, 28624}, {8887, 19169}, {9076, 32085}, {9308, 35259}, {10101, 37965}, {10282, 14363}, {10301, 16264}, {10425, 14221}, {11005, 12828}, {11634, 20187}, {12032, 31905}, {12121, 32418}, {13195, 17409}, {13395, 18026}, {13398, 30512}, {13597, 34484}, {14006, 29056}, {14013, 28848}, {14185, 30252}, {14187, 30253}, {14615, 15259}, {14978, 18369}, {14979, 16337}, {15149, 28838}, {15440, 32676}, {16868, 22751}, {18809, 32417}, {20189, 36829}, {21396, 37814}, {23591, 23976}, {23701, 37855}, {23999, 36066}, {24000, 36069}, {24021, 36068}, {26714, 35325}, {28145, 31902}, {28173, 31900}, {28193, 31903}, {28197, 31901}, {28842, 31904}, {28844, 31909}, {28857, 31914}, {28861, 31907}, {28884, 31908}, {28892, 31921}, {32681, 32695}, {34473, 36176}, {36079, 36118}, {37760, 37766}

▸ Functional Relations27 relations

• X₁₀₇ = center of the bianticevian conic of X₁ and X₄, the rectangular hyperbola passing through X₁, X₄, X₁₉, and the vertices of their anticevian triangles. This hyperbola is the excentral isogonal conjugate of line X₄₀X₂₉₃₉, the anticomplementary conjugate of line X₂₀X₁₃₃₀, and the anticomplementary isotomic conjugate of line X₁₆₅₄X₃₁₆₄. (Randy Hutson, April 9, 2016)

• X₁₀₇ = reflection of X_i in X_j for these (i,j): (4, 133), (1294, 3)

• X₁₀₇ = isogonal conjugate of X₅₂₀

• X₁₀₇ = isotomic conjugate of X₃₂₆₅

• X₁₀₇ = complement of X₃₄₁₈₆

• X₁₀₇ = anticomplement of X₁₂₂

• X₁₀₇ = cevapoint of X₄ and X₅₂₃

• X₁₀₇ = cevapoint of X₂₄₀₀₇ and X₂₄₀₀₈ (the Kiepert hyperbola intercepts of the orthic axis)

• X₁₀₇ = X_i-cross conjugate of X_j for these (i,j): (24, 250), (108, 162), (523, 4)

• X₁₀₇ = trilinear pole of line X₄X₆

• X₁₀₇ = Ψ(X_i,X_j) for these (i,j): (1, 29), (3, 2), (6, 4), (4, 51), (54, 4), (64, 4), (65, 4), (67, 4), (69, 4)

• X₁₀₇ = intersection of reflections in sides of ABC of line X₄X₅₁

• X₁₀₇ = reflection of X₁₃₀₄ in the Euler line

• X₁₀₇ = reflection of X₂₇₁₃ in the Brocard axis

• X₁₀₇ = reflection of X₂₇₁₉ in line X₁X₃

• X₁₀₇ = inverse-in-polar-circle of X₁₂₅

• X₁₀₇ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₃₂

• X₁₀₇ = pole wrt polar circle of trilinear polar of X₅₂₅ (line X₁₂₂X₁₂₅)

• X₁₀₇ = X₄₈-isoconjugate (polar conjugate) of X₅₂₅

• X₁₀₇ = X₁₅₇₇-isoconjugate of X₅₇₇

• X₁₀₇ = crossdifference of every pair of points on line X₁₆₃₆X₂₉₇₂

• X₁₀₇ = X₁₃₄-of-excentral-triangle

• X₁₀₇ = circumcircle intercept, other than A, B, C, of conic {A, B, C,PU(157)}}

• X₁₀₇ = Thomson-isogonal conjugate of X₆₀₀₀

• X₁₀₇ = Lucas-isogonal conjugate of X₆₀₀₀

• X₁₀₇ = barycentric product of Steiner circumellipse intercepts of van Aubel line

• X₁₀₇ = trilinear product of circumcircle intercepts of line X₁X₂₉