X₁₃₁₃

2nd MOSES INTERSECTION

X₁₃₁₃ = midpoint of X₄ and X₁₁₁₄

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

X₁₃₁₃ (2nd MOSES INTERSECTION) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 1114}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₁₁₁₄ (2nd EULER-LINE-CIRCUMCIRCLE INTERSECTION), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \((J + 1)\cos A + 2(J - 1)\cos B \cos C\), where \(J = |OH|/R\); see X₁₁₁₃

Barycentrics: (J + 1)\(a^{2}\)SA + 2(J - 1) SB SC; cf. X₁₃₁₂

Search Value (6, 9, 13): \(3.748515527836\)

▸ Linear Combinations1 combination

• X₁₃₁₃ = \(3\) X₂ + \((-3 + |OH|/R)\) X₅ = \(3(1 - |OH|/R)\) X₂ + \((-3 + |OH|/R)\) X₃

X₁₃₁₃ lies on the nine-point circle, the MacBeath inconic.

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations8 relations

• X₁₃₁₃ = midpoint of X₄ and X₁₁₁₄

• X₁₃₁₃ = reflection of X₁₃₁₂ in X₅

• X₁₃₁₃ = complement of X₁₁₁₃

• X₁₃₁₃ = X₁₁₁₄-Ceva conjugate of X₅₂₃

• X₁₃₁₃ = inverse-in-polar-circle of X₁₁₁₃

• X₁₃₁₃ = X₁₃₈₂-of-orthic-triangle if ABC is acute

• X₁₃₁₃ = excentral-to-ABC functional image of X₁₃₈₂

• X₁₃₁₃ = {X₂,X₁₁₁₄}-harmonic conjugate of X₄₆₈

Notes

As a point on the Euler line, X₁₃₁₃ has Shinagawa coefficients \((J + 1, J - 3)\).

Of the points other than X₁₂₅ in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X₁₃₁₃ is the one nearer to X₃. (Randy Hutson, December 2, 2017)