X₄₇₉

X₂₆₉-CROSS CONJUGATE OF X₂₇₉

Let A' be the point in which the incircle is tangent to a circle that passes through vertices B and C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₄₇₉. For an excircle version, see X₅₄₂₃.

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

X₄₇₉ (X₂₆₉-CROSS CONJUGATE OF X₂₇₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {57, 279}, {7, 354}

• Related Centers: X₂₆₉ (ISOGONAL CONJUGATE OF X₂₀₀), X₂₇₉ (ISOGONAL CONJUGATE OF X₂₂₀), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₇ (GERGONNE POINT), X₃₅₄ (WEILL POINT)

Trilinears: \((\tan A/2 \sec A/2)^{2} : :\)

Barycentrics: \(\tan^{3}(A/2) : :\)

Search Value (6, 9, 13): \(0.015254626563\)

▸ Alternate Coordinates1 form

• Barycentrics: \(1/(b + c - a)^{3} : :\)

▸ Incident Central Lines16 lines

{2, 5574}, {7, 354}, {8, 7182}, {55, 3160}, {57, 279}, {165, 1323}, {269, 614}, {348, 5273}, {658, 5435}, {934, 1617}, {1014, 5324}, {1119, 1851}, {1407, 1462}, {1996, 5226}, {3188, 6060}, {4554, 8055}

▸ Functional Relations10 relations

• X₄₇₉ = isogonal conjugate of X₄₈₀

• X₄₇₉ = isotomic conjugate of X₅₄₂₃

• X₄₇₉ = X₂₆₉-cross conjugate of X₂₇₉

• X₄₇₉ = anticomplement of X₅₅₇₄

• X₄₇₉ = crosssum of X₃₀₂₂ and X₄₁₀₅

• X₄₇₉ = cevapoint of X_i and X_j for these {i,j}: {269, 738}, {3271, 3669}

• X₄₇₉ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (1088, 7056, 7), (3160, 3599, 55)

• X₄₇₉ = X_i-cross conguate of X_j for these (i,j): (269, 279), (3271, 3669), (5573, 2), (7195, 7)

• X₄₇₉ = polar conjugate of isotomic conjugate of X₃₀₆₈₂

• X₄₇₉ = X_i-isoconjugate of X_j for these {i,j}: {1, 480}, {2, 6602}, {6, 728}, {8, 1253}, {9, 220}, {31, 5423}, {33, 1260}, {41, 346}, {55, 200}, {78, 7071}, {100, 4105}, {101, 4130}, {210, 2328}, {212, 7046}, {219, 7079}, {281, 1802}, {282, 7368}, {284, 4515}, {341, 2175}, {607, 3692}, {643, 4524}, {644, 657}, {663, 4578}, {692, 4163}, {756, 6061}, {765, 3022}, {1098, 7064}, {1110, 4081}, {1252, 3119}, {1265, 2212}, {1334, 2287}, {2194, 4082}, {2310, 6065}, {2318, 4183}, {2324, 7367}, {2332, 3694}, {3063, 6558}, {3699, 8641}, {3709, 7259}, {3900, 3939}, {4012, 7084}, {4171, 5546}, {6605, 8012}

Notes

Clark Kimberling and Peter Yff, Problem 10678, American Mathematical Monthly 105 (1998) 666.