X₅₅₉

(X₁,X₁₅)-ANSWER TO QUESTION A

X₅₅₉ = isogonal conjugate of X₃₃₆₅₃

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

X₅₅₉ ((X₁,X₁₅)-ANSWER TO QUESTION A) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {14, 226}

• Related Centers: X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₄ (2nd 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): \(1.229920978133\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a ((a + b - c) \cdot (a - b + c) + 2 \sqrt{3} S) : :\)

X₅₅₉ lies on the cubics K134 and K341b.

▸ Incident Central Lines40 lines

{1, 3}, {2, 5239}, {6, 7088}, {7, 1081}, {14, 226}, {16, 16577}, {63, 5240}, {81, 7052}, {222, 7059}, {299, 319}, {465, 17043}, {497, 37830}, {553, 3639}, {651, 7126}, {1100, 1652}, {1255, 33654}, {1277, 21475}, {1442, 10638}, {1653, 16777}, {1836, 10652}, {1962, 10647}, {2003, 5357}, {2306, 25417}, {3219, 11126}, {3474, 37833}, {3475, 30344}, {4336, 30301}, {7005, 42677}, {7051, 17011}, {7127, 37787}, {7202, 42624}, {9778, 30338}, {10391, 10650}, {10580, 30339}, {10648, 17017}, {10651, 11246}, {14100, 30357}, {16038, 17718}, {17778, 37794}, {19373, 28606}

▸ Functional Relations8 relations

• X₅₅₉ = isogonal conjugate of X₃₃₆₅₃

• X₅₅₉ = X₇₃₄₄-complementary conjugate of X₁₄₁

• X₅₅₉ = X₁₂₅₅-Ceva conjugate of X₁₀₈₂

• X₅₅₉ = X_i-Dao conjugate of X_j for these (i,j): {3, 33653}, {223, 554}, {478, 33654}, {5239, 40581}

• X₅₅₉ = X_i-isoconjugate of X_j for these (i,j): {1, 33653}, {9, 33654}, {14, 7127}, {55, 554}, {79, 1250}, {522, 36073}, {1082, 7073}, {1251, 42680}, {2154, 5239}, {2307, 7110}, {5240, 11073}, {6186, 40713}, {7126, 41225}, {11086, 36933}, {19551, 39152}

• X₅₅₉ = barycentric product X_i*X_j for these {i,j}: {57, 40714}, {85, 10638}, {299, 7052}, {319, 2306}, {1081, 3219}, {1251, 17095}, {18160, 36072}, {33654, 46176}

• X₅₅₉ = barycentric quotient X_i/X_j for these {i,j}: {6, 33653}, {16, 5239}, {56, 33654}, {57, 554}, {1081, 30690}, {1251, 7110}, {1399, 2307}, {1415, 36073}, {2003, 1082}, {2152, 7127}, {2174, 1250}, {2306, 79}, {2307, 42680}, {3219, 40713}, {5357, 5240}, {7043, 44691}, {7051, 41225}, {7052, 14}, {7150, 7043}, {10638, 9}, {19373, 39152}, {39150, 36932}, {40714, 312}, {42624, 7126}, {46073, 7026}

• X₅₅₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 57, 1082}, {1, 37773, 37772}, {56, 20182, 1082}, {57, 1082, 37772}, {65, 37595, 1082}, {226, 3638, 554}, {241, 3748, 1082}, {940, 2099, 1082}, {1082, 37773, 57}, {1214, 24929, 1082}, {1319, 3666, 1082}, {1429, 17598, 1082}, {7146, 17716, 1082}, {13388, 13389, 37773}, {15934, 37543, 1082}