X₁₀₃

ANTIPODE OF X₁₀₁

X₁₀₃ is the antipode of X₁₀₁.

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

X₁₀₃ (ANTIPODE OF X₁₀₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 150}

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₁₅₀ (REFLECTION OF X₂₀ IN X₁₀₃)

Trilinears: \(a/[(a - b) \cot C + (a - c) \cot B] : b/[(b - c) \cot A + (b - a) \cot C] : c/[(c - a) \cot B + (c - b) \cot A]\)

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

Tripolars: \(|b c ((a - b) \cot C + (a - c) \cot B)| : :\)

Search Value (6, 9, 13): \(12.244219052041\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2} / (2 a^{3} - a^{2} (b + c) - (b - c)^{2} (b + c)) : :\)

▸ Incident Central Lines164 lines

{1, 934}, {2, 118}, {3, 101}, {4, 116}, {5, 10741}, {6, 10758}, {11, 10772}, {20, 150}, {27, 107}, {30, 2690}, {33, 57}, {35, 1803}, {36, 4845}, {40, 1292}, {46, 18413}, {55, 109}, {56, 3022}, {58, 112}, {63, 100}, {64, 4306}, {74, 2774}, {84, 7719}, {98, 2786}, {99, 1043}, {102, 928}, {104, 3887}, {105, 2254}, {106, 2424}, {110, 1790}, {111, 2824}, {140, 38764}, {182, 30554}, {295, 813}, {354, 7073}, {376, 544}, {381, 38768}, {386, 34457}, {476, 5196}, {477, 37166}, {511, 2702}, {512, 2700}, {513, 2717}, {514, 2724}, {515, 929}, {516, 927}, {517, 1308}, {518, 2736}, {519, 2737}, {520, 2738}, {521, 2739}, {522, 2723}, {523, 2688}, {524, 2740}, {525, 2741}, {549, 38774}, {550, 38766}, {572, 825}, {573, 3033}, {580, 29041}, {631, 6710}, {654, 2291}, {672, 919}, {675, 2400}, {677, 901}, {789, 24728}, {831, 9943}, {835, 34255}, {910, 971}, {916, 15380}, {1146, 18328}, {1147, 3046}, {1155, 2222}, {1290, 5536}, {1293, 1350}, {1294, 2811}, {1295, 2812}, {1296, 2813}, {1297, 9518}, {1302, 7474}, {1304, 2073}, {1309, 34234}, {1310, 12520}, {1326, 2715}, {1376, 3041}, {1458, 24016}, {1461, 3270}, {1499, 2729}, {1593, 5185}, {1617, 2192}, {1656, 38767}, {1742, 7220}, {1764, 38479}, {1796, 8701}, {1810, 6078}, {2077, 2742}, {2078, 2342}, {2093, 34930}, {2221, 32691}, {2272, 3220}, {2280, 28899}, {2705, 18860}, {2716, 7634}, {2725, 3309}, {2726, 3667}, {2727, 6000}, {2728, 6001}, {3090, 38769}, {3190, 35987}, {3333, 14760}, {3428, 28291}, {3430, 6010}, {3522, 20096}, {3523, 38772}, {3525, 20401}, {3533, 38770}, {3576, 11712}, {3587, 34925}, {3616, 11728}, {3732, 31852}, {4256, 32722}, {4257, 26715}, {4300, 29279}, {5010, 34931}, {5188, 28486}, {5542, 10136}, {5584, 6575}, {5603, 11726}, {5840, 10770}, {6011, 30271}, {6135, 6213}, {6136, 6212}, {6282, 30237}, {6577, 15622}, {7072, 36082}, {7465, 9058}, {7466, 9107}, {7688, 20219}, {8608, 32684}, {8722, 28564}, {8750, 22084}, {9085, 35365}, {9320, 12032}, {9441, 9503}, {10164, 28346}, {10246, 32630}, {10299, 35024}, {10303, 38775}, {10884, 13395}, {11825, 34112}, {13397, 20243}, {13444, 16012}, {13478, 26704}, {15730, 30282}, {17729, 36028}, {21153, 28345}, {28162, 36942}, {28469, 30270}, {28474, 30269}, {28841, 37508}, {29044, 37469}, {29055, 37575}, {29091, 31732}, {29217, 37482}, {32642, 32682}

▸ Functional Relations27 relations

• X₁₀₃ = midpoint of X₂₀ and X₁₅₀

• X₁₀₃ = reflection of X_i in X_j for these (i,j): (4, 116), (101, 3), (152, 118)

• X₁₀₃ = isogonal conjugate of X₅₁₆

• X₁₀₃ = isotomic conjugate of X₃₅₅₁₇

• X₁₀₃ = complement of X₁₅₂

• X₁₀₃ = anticomplement of X₁₁₈

• X₁₀₃ = X₂₁-beth conjugate of X₉₃₄

• X₁₀₃ = circumcircle-antipode of X₁₀₁

• X₁₀₃ = X₁₁₅-of-the-hexyl-triangle

• X₁₀₃ = perspector of ABC and the triangle formed by reflecting line PU(10) in the sidelines of ABC

• X₁₀₃ = X₁₁₄-of-excentral-triangle

• X₁₀₃ = trilinear pole of line X₆X₆₅₇

• X₁₀₃ = Ψ(X_i,X_j) for these (i,j): (6, 657), (101, 3), (190, 69)

• X₁₀₃ = Λ(X₁, X₇)

• X₁₀₃ = trilinear pole wrt 2nd circumperp triangle of line X₁₀₀₁X₁₀₁₂

• X₁₀₃ = X₉₉-of-2nd-circumperp-triangle

• X₁₀₃ = reflection of X₂₆₈₈ in the Euler line

• X₁₀₃ = reflection of X₂₇₀₀ in the Brocard axis

• X₁₀₃ = reflection of X₂₇₁₇ in line X₁X₃

• X₁₀₃ = Thomson-isogonal conjugate of X₅₁₄

• X₁₀₃ = Lucas-isogonal conjugate of X₅₁₄

• X₁₀₃ = Cundy-Parry Phi transform of X₃₇₃₀

• X₁₀₃ = Cundy-Parry Psi transform of X₁₄₃₇₇

• X₁₀₃ = polar conjugate of isogonal conjugate of X₃₂₆₅₇

• X₁₀₃ = areal center of pedal triangles of PU(10)

• X₁₀₃ = X₁₉-isoconjugate of X₂₆₀₀₆

• X₁₀₃ = SR(P,U), where P and U are the circumcircle intercepts of the Soddy line

Notes

Let J_aJ_bJ_c be the excentral triangle. The Lemoine axes of triangles J_aBC, J_bCA, J_cAB bound a triangle perspective to ABC at X₁₀₃. (Randy Hutson, June 27, 2018)