X₄₅₈

EULER LINE INTERCEPT OF LINE X₇₆X_275

X₄₅₈ = inverse-in-orthocentroidal-circle of X₂₉₇

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

X₄₅₈ (EULER LINE INTERCEPT OF LINE X₇₆X_275) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 264}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₂₆₄ (ISOTOMIC CONJUGATE OF CIRCUMCENTER)

Trilinears: \(u \csc 2A : v \csc 2B : w \csc 2C\), where \(u : v : w\) = X₁₈₂; e.g.; \(u(A,B,C) = \cos(A - \omega )\)

Barycentrics: \(u \sec A : v \sec B : w \sec C\)

Search Value (6, 9, 13): \(3.233562950894\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\tan A + \cot \omega : :\)

• Barycentrics: \(\sec A \cos(A - \omega ) : :\)

▸ Incident Central Lines8 lines

{2, 3}, {6, 264}, {76, 275}, {141, 317}, {239, 318}, {273, 894}, {315, 343}, {340, 599}

▸ Functional Relations3 relations

• X₄₅₈ = inverse-in-orthocentroidal-circle of X₂₉₇

• X₄₅₈ = pole wrt polar circle of trilinear polar of X₂₆₂ (line X₅₂₃X₃₅₆₉)

• X₄₅₈ = polar conjugate of X₂₆₂

Notes

As a point on the Euler line, X₄₅₈ has Shinagawa coefficients \(((E + F)F, S^{2})\).