X₂₄₇₅

ANTICOMPLEMENT OF X₂₁

X₂₄₇₅ is the anticomplement of X₂₁.

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

X₂₄₇₅ (ANTICOMPLEMENT OF X₂₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 21}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₂₁ (SCHIFFLER POINT), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[a^{4} + a^{2}bc + abc(b + c) - (b^{2} - c^{2})^{2}]\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.533772656836\)

▸ Linear Combinations1 combination

• X₂₄₇₅ = \(3\) X₂ - \(2\) X₂₁

▸ Incident Central Lines76 lines

{1, 149}, {2, 3}, {7, 2893}, {8, 79}, {10, 191}, {11, 5253}, {12, 100}, {35, 3822}, {65, 5086}, {75, 5016}, {80, 3754}, {81, 1834}, {115, 5277}, {144, 1654}, {145, 388}, {148, 1655}, {153, 355}, {193, 5800}, {214, 5443}, {225, 4296}, {274, 316}, {323, 3193}, {495, 3871}, {497, 3622}, {518, 5178}, {519, 5270}, {535, 5258}, {551, 4857}, {950, 5249}, {960, 5057}, {966, 5036}, {1030, 5949}, {1043, 3936}, {1056, 3623}, {1125, 3583}, {1158, 5587}, {1220, 4972}, {1441, 2897}, {1448, 2000}, {1479, 3616}, {1621, 6284}, {1698, 4333}, {1836, 3869}, {1837, 4459}, {1869, 3101}, {1901, 2287}, {1993, 5706}, {2646, 3838}, {2794, 5985}, {2886, 2975}, {2894, 3419}, {3035, 3614}, {3218, 4292}, {3294, 5134}, {3421, 4678}, {3621, 5082}, {3652, 5818}, {3679, 3951}, {3770, 4696}, {3813, 5434}, {3841, 5251}, {3870, 5290}, {3878, 5180}, {3909, 5252}, {3925, 5260}, {4316, 5267}, {4385, 5300}, {4652, 5705}, {4855, 5219}, {4968, 5015}, {4999, 5303}, {5176, 5836}, {5254, 5276}, {5318, 5362}, {5321, 5367}, {5554, 6223}

▸ Functional Relations8 relations

• X₂₄₇₅ = reflection of X_i in X_j for these (i,j): (21, 442), (191, 10)

• X₂₄₇₅ = isogonal conjugate of X₃₄₄₃₅

• X₂₄₇₅ = inverse-in-Fuhrmann circle of X₃₄₄₈

• X₂₄₇₅ = anticomplement of X₂₁

• X₂₄₇₅ = X₁₄₄₁-Ceva conjugate of X₂

• X₂₄₇₅ = X₅₄ of Fuhrmann triangle

• X₂₄₇₅ = {X₂,X₂₁}-harmonic conjugate of X₁₅₆₇₄

• X₂₄₇₅ = crosspoint of X₄ and X₈ wrt extraversion triangle of X₈

Notes

As a point on the Euler line, X₂₄₇₅ has Shinagawa coefficients \((abc\$a\$, 4S^{2}), and also Shinagawa coefficients (R, 2r).\).

Let F be the Fuhrmann triangle. Then F is perspective to F-of-orthic-triangle-of-F, and the perspector is X₂₄₇₅. (Randy Hutson, June 15, 2015)