X₁₅₆₇

ORTHOJOIN OF X₆₉₄

X₁₅₆₇ = perspector of circumconic centered at X₆₉₄

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

X₁₅₆₇ (ORTHOJOIN OF X₆₉₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X₁₅₁₂)

Search Value (6, 9, 13): \(-5.358096960804\)

X₁₅₆₇ lies on the nine-point circle.

▸ Incident Central Lines1 line

{2, 3}

▸ Functional Relations3 relations

• X₁₅₆₇ = perspector of circumconic centered at X₆₉₄

• X₁₅₆₇ = center of circumconic that is locus of trilinear poles of lines passing through X₆₉₄

• X₁₅₆₇ = X₂-Ceva conjugate of X₆₉₄

Notes

As a point on the Euler line, X₁₅₆₇ has Shinagawa coefficients \(((E + F)^{6} - (5E - 3F)(E + F)^{3}S^{2} + 3(E + F)^{2}S^{4} + S^{6}, -(E + 3F)(E + F)^{5} + 3(E - 3F)(E + F)^{3}S^{2} + 9(E - F)(E + F)S^{4} - 3S^{6})\).