X₉₀₇

POINT ACRUX

X₉₀₇ = isogonal conjugate of X₃₈₀₀

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

X₉₀₇ (POINT ACRUX) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {98, 620}, {111, 1180}

• Related Centers: X₃₈₀₀ (com[INVERSE(nT(a2, a2 + b2))]), X₉₈ (TARRY POINT), X₆₂₀ (MIDPOINT OF X₆₁₈ AND X₆₁₉), X₁₁₁ (PARRY POINT), X₁₁₈₀ (2nd SARAGOSSA POINT OF X₂)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a/[(b^{2} - c^{2})(3a^{2} + b^{2} + c^{2})]\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}/[(b^{2} - c^{2})(3a^{2} + b^{2} + c^{2})]\)

Search Value (6, 9, 13): \(1.553625773911\)

X₉₀₇ lies on the circumcircle.

▸ Incident Central Lines3 lines

{98, 620}, {111, 1180}, {112, 1634}

▸ Functional Relations2 relations

• X₉₀₇ = isogonal conjugate of X₃₈₀₀

• X₉₀₇ = Λ(X₄₁₉, X₂₅₀₁) (line is radical axis of circumcircle and orthosymmedial circle)