X₁₀₈₂

POINT ALPHERATZ

Coordinates for X₁₀₈₂ are obtained from those of X₅₅₉ by changing π/3 to - π/3; contributed by Edward Brisse.

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

X₁₀₈₂ (POINT ALPHERATZ) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {13, 226}

• Related Centers: X₁₂₅₁ (ISOGONAL CONJUGATE OF X₁₀₈₂), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₃ (1st ISOGONIC CENTER), X₂₂₆ (-CEVA CONJUGATE OF X₆₅)

Trilinears: \((\sec A/2) \sin(A/2 - \pi /3) : (\sec B/2) \sin(B/2 - \pi /3) : (\sec C/2) \sin(C/2 - \pi /3)\)

Barycentrics: \((\sin A/2) \sin(A/2 - \pi /3) : (\sin B/2) \sin(B/2 - \pi /3) : (\sin C/2) \sin(C/2 - \pi /3)\)

Search Value (6, 9, 13): \(3.240228702104\)

X₁₀₈₂ lies on the cubics K134 and K341a.

▸ Incident Central Lines37 lines

{1, 3}, {2, 5240}, {6, 7089}, {7, 554}, {13, 226}, {15, 16577}, {63, 5239}, {81, 33655}, {222, 7060}, {298, 319}, {466, 17043}, {497, 37833}, {553, 3638}, {651, 19551}, {1100, 1653}, {1250, 1442}, {1255, 2306}, {1276, 21476}, {1652, 16777}, {1836, 10651}, {1962, 10648}, {2003, 5353}, {2307, 3219}, {3474, 37830}, {3475, 30345}, {4336, 30300}, {7006, 42680}, {7051, 28606}, {9778, 30339}, {10391, 10649}, {10580, 30338}, {10647, 17017}, {10652, 11246}, {14100, 30356}, {17011, 19373}, {17778, 37795}, {25417, 33654}

▸ Functional Relations9 relations

• X₁₀₈₂ = isogonal conjugate of X₁₂₅₁

• X₁₀₈₂ = X₇₃₄₅-complementary conjugate of X₁₄₁

• X₁₀₈₂ = X₁₂₅₅-Ceva conjugate of X₅₅₉

• X₁₀₈₂ = cevapoint of X₁₂₅₀ and X₂₃₀₇

• X₁₀₈₂ = barycentric product X_i*X_j for these {i,j}: {57, 40713}, {75, 2307}, {85, 1250}, {298, 33655}, {319, 33654}, {554, 3219}, {17095, 33653}, {18160, 36073}

• X₁₀₈₂ = X_i-Dao conjugate of X_j for these (i,j): {3, 1251}, {223, 1081}, {478, 2306}, {5240, 40580}

• X₁₀₈₂ = X_i-isoconjugate of X_j for these (i,j): {1, 1251}, {9, 2306}, {55, 1081}, {79, 10638}, {522, 36072}, {559, 7073}, {2153, 5240}, {3179, 19551}, {5239, 11072}, {6186, 40714}, {7026, 42623}, {7126, 39153}, {11081, 36932}, {33653, 42677}

• X₁₀₈₂ = barycentric quotient X_i/X_j for these {i,j}: {6, 1251}, {15, 5240}, {56, 2306}, {57, 1081}, {554, 30690}, {1250, 9}, {1415, 36072}, {2003, 559}, {2174, 10638}, {2307, 1}, {3219, 40714}, {5353, 5239}, {7026, 44690}, {7051, 39153}, {19373, 3179}, {33653, 7110}, {33654, 79}, {33655, 13}, {39151, 36933}, {40713, 312}, {46077, 7043}

• X₁₀₈₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 57, 559}, {1, 37772, 37773}, {56, 20182, 559}, {57, 559, 37773}, {65, 37595, 559}, {226, 3639, 1081}, {241, 3748, 559}, {559, 37772, 57}, {940, 2099, 559}, {1214, 24929, 559}, {1319, 3666, 559}, {1429, 17598, 559}, {7146, 17716, 559}, {13388, 13389, 37772}, {15934, 37543, 559}