X₁₀₈₁

POINT ALPHECCA

Coordinates for X₁₀₈₁ are obtained from those of X₅₅₄ by changing π/3 to - π/3; contributed by Edward Brisse.

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

X₁₀₈₁ (POINT ALPHECCA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 30}, {13, 226}

• Related Centers: X₁₂₅₀ (ISOGONAL CONJUGATE OF X₁₀₈₁), X₁ (INCENTER), X₃₀ (EULER INFINITY POINT), X₁₃ (1st ISOGONIC CENTER), X₂₂₆ (-CEVA CONJUGATE OF X₆₅)

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

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

Search Value (6, 9, 13): \(0.026811797395\)

▸ Alternate Coordinates1 form

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

X₁₀₈₁ lies on the cubics K134 and K419a.

▸ Incident Central Lines23 lines

{1, 30}, {2, 2306}, {7, 559}, {13, 226}, {14, 43682}, {55, 10651}, {57, 3179}, {75, 298}, {395, 1653}, {497, 30345}, {553, 37773}, {675, 36072}, {1086, 11072}, {1365, 18974}, {2153, 41889}, {3475, 37833}, {3638, 3982}, {5239, 5905}, {5240, 5249}, {6186, 10647}, {7026, 11078}, {11706, 26700}, {30328, 37640}

▸ Functional Relations9 relations

• X₁₀₈₁ = isogonal conjugate of X₁₂₅₀

• X₁₀₈₁ = isotomic conjugate of X₄₀₇₁₃

• X₁₀₈₁ = X_i-cross conjugate of X_j for these (i,j): {553, 554}, {30383, 85}

• X₁₀₈₁ = cevapoint of X_i and X_j for these (i,j): {1, 1653}, {1251, 2306}

• X₁₀₈₁ = X_i-Dao conjugate of X_j for these (i,j): {2, 40713}, {3, 1250}, {223, 1082}, {478, 2307}, {7026, 40578}

• X₁₀₈₁ = X_i-isoconjugate of X_j for these (i,j): {1, 1250}, {9, 2307}, {15, 19551}, {31, 40713}, {35, 33653}, {55, 1082}, {1251, 7006}, {2151, 7026}, {5353, 7126}, {7127, 46077}, {10638, 42680}, {35057, 36073}

• X₁₀₈₁ = barycentric product X_i*X_j for these {i,j}: {75, 2306}, {85, 1251}, {300, 19373}, {559, 30690}, {3261, 36072}

• X₁₀₈₁ = barycentric quotient X_i/X_j for these {i,j}: {2, 40713}, {6, 1250}, {13, 7026}, {56, 2307}, {57, 1082}, {559, 3219}, {1251, 9}, {2153, 19551}, {2160, 33653}, {2306, 1}, {2307, 7006}, {3179, 5240}, {5240, 44688}, {7051, 5353}, {7052, 46077}, {11072, 7126}, {19373, 15}, {33654, 42680}, {36072, 101}, {39153, 5239}, {40714, 42033}

• X₁₀₈₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {1, 4654, 554}, {226, 3639, 1082}, {3649, 37631, 554}, {3782, 5434, 554}