X₁₆₅₀

TRIPOLAR CENTROID OF X₅₂₅

X₁₆₅₀ is the tripolar centroid of X₅₂₅.

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

X₁₆₅₀ (TRIPOLAR CENTROID OF X₅₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1651, 2}, {2, 3} (Euler line)

• Related Centers: X₁₆₅₁ (TRIPOLAR CENTROID OF X₆₄₈), X₂ (CENTROID), X₃ (CIRCUMCENTER)

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

Search Value (6, 9, 13): \(3.740536426287\)

▸ Incident Central Lines2 lines

{2, 3}, {122, 125}

▸ Functional Relations8 relations

• X₁₆₅₀ = reflection of X₁₆₅₁ in X₂

• X₁₆₅₀ = anticomplement of X₄₀₂

• X₁₆₅₀ = nine-point-circle-inverse of X₃₇₉₈₅

• X₁₆₅₀ = X_i-Ceva conjugate of X_j for these (i,j): (265, 520), (1294, 523), (1494, 525)

• X₁₆₅₀ = crosspoint of X₅₂₅ and X₁₄₉₄

• X₁₆₅₀ = crosssum of X_i and X_j for these (i,j): (74, 1304), (110, 2071), (112, 1495)

• X₁₆₅₀ = complement of X₄₂₄₀

• X₁₆₅₀ = homothetic center of Gossard and medial triangles

Notes

As a point on the Euler line, X₁₆₅₀ has Shinagawa coefficients \(((4E - 5F)F - S^{2}, -3(E + F)F + S^{2})\).

Let W be the circumconic with center X₁₆₅₀. One of the asymptotes of W is the Euler line. The other is in the direction of X₉₀₃₃. For a sketch, click X₉₀₃₃ (X(9033).gif). (Angel Montesdeoca, April 19, 2016)