X₄₆₀

POINT ANTARES

X₄₆₀ = X₂₄₁-of-orthic-triangle if ABC is acute

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

X₄₆₀ (POINT ANTARES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {53, 1974}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₃ (SYMMEDIAN POINT OF ORTHIC TRIANGLE), X₁₉₇₄ (BARYCENTRIC PRODUCT OF PU(39))

Trilinears: \((\sec A)[a^{2}(2a^{2} - b^{2} - c^{2}) + (b^{2} - c^{2})^{2}] : :\)

Search Value (6, 9, 13): \(-0.077383014685\)

▸ Incident Central Lines4 lines

{2, 3}, {53, 1974}, {512, 2501}, {685, 2065}

▸ Functional Relations5 relations

• X₄₆₀ = crossdifference of every pair of points on line X₃₉₄X₆₄₇

• X₄₆₀ = X₂₄₁-of-orthic-triangle if ABC is acute

• X₄₆₀ = pole wrt polar circle of trilinear polar of X₈₇₈₁ (line X₆₉X₅₂₃)

• X₄₆₀ = polar conjugate of X₈₇₈₁

• X₄₆₀ = barycentric product X₄*X₂₃₀

Notes

As a point on the Euler line, X₄₆₀ has Shinagawa coefficients \(((E + F)F, -(E + F)^{2} + 2S^{2})\).