X₉₀₈

POINT ACUBENS

X₉₀₈ = crosssum of X₄₁ and X₉₀₂

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

X₉₀₈ (POINT ACUBENS) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1512, 119}

• Related Centers: X₁₅₁₂ (ORTHOJOIN OF X₁), X₁₁₉ (FEUERBACH ANTIPODE)

Trilinears: \([(wb + vc)/a - v - w]/a\), where \(u : v : w\) = X₃

Barycentrics: \(\cos B + \cos C - 1 : :\)

Search Value (6, 9, 13): \(9.953055690939\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a^{2} (b + c) - 2 a b c - (b - c)^{2} (b + c) : :\)

▸ Incident Central Lines18 lines

{1, 998}, {2, 7}, {4, 78}, {5, 72}, {8, 946}, {10, 994}, {11, 518}, {12, 960}, {80, 519}, {92, 264}, {100, 516}, {119, 517}, {153, 515}, {214, 535}, {224, 1079}, {377, 936}, {392, 495}, {514, 661}

▸ Functional Relations13 relations

• X₉₀₈ = reflection of X₁₅₁₂ in X₁₁₉

• X₉₀₈ = isogonal conjugate of X₉₀₉

• X₉₀₈ = isotomic conjugate of X₃₄₂₃₄

• X₉₀₈ = anticomplement of X₃₉₁₁

• X₉₀₈ = complement of X₃₂₁₈

• X₉₀₈ = crosssum of X₄₁ and X₉₀₂

• X₉₀₈ = crossdifference of every pair of points on line X₃₁X₆₆₃

• X₉₀₈ = inverse-in-circumconic-centered-at-X₉ of X₆₃

• X₉₀₈ = trilinear pole of line X₁₁₄₅X₁₇₆₉

• X₉₀₈ = {X₂,X₇}-harmonic conjugate of X₃₃₀₆

• X₉₀₈ = polar conjugate of X₃₆₁₂₃

• X₉₀₈ = pole wrt polar circle of trilinear polar of X₃₆₁₂₃ (line X₁₉X₆₄₉)

• X₉₀₈ = homothetic center of the complement of the excentral triangle and the anticomplement of the intouch triangle

Notes

Let L_a, L_b, L_c be the lines through A, B, C, resp. parallel to the antiorthic axis. Let M_a, M_b, M_c be the reflections of BC, CA, AB in L_a, L_b, L_c, resp. Let A' = M_b∩M_c, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A″B″C″ be the reflection of A'B'C' in the antiorthic axis. The triangle A″B″C″ is homothetic to ABC, with center of homothety X₉₀₈; see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)