X₁₆₅₁

TRIPOLAR CENTROID OF X₆₄₈

X₁₆₅₁ is the tripolar centroid of X₆₄₈.

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

X₁₆₅₁ (TRIPOLAR CENTROID OF X₆₄₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1636, 1637}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₆₃₆ (TRIPOLAR CENTROID OF X₃), X₁₆₃₇ (TRIPOLAR CENTROID OF X₄)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X₁₆₃₅)

Search Value (6, 9, 13): \(1.518201158690\)

▸ Incident Central Lines2 lines

{2, 3}, {1636, 1637}

▸ Functional Relations4 relations

• X₁₆₅₁ = reflection of X_i in X_j for these (i,j): (2, 402), (1650, 2)

• X₁₆₅₁ = Euler line intercept of trilinear polar of X₃₀

• X₁₆₅₁ = centroid of Euler line intercepts with sidelines of ABC

• X₁₆₅₁ = X₂-of-Gossard triangle

Notes

As a point on the Euler line, X₁₆₅₁ has Shinagawa coefficients \((3(2E - 7F)F - S^{2},9(E + F)F - 3S^{2})\).