X₉₅

CEVAPOINT OF CENTROID AND CIRCUMCENTER

X₉₅ is the cevapoint of centroid and circumcenter.

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

X₉₅ (CEVAPOINT OF CENTROID AND CIRCUMCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {54, 69}, {2, 97}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₅₄ (KOSNITA POINT), X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE), X₉₇ (ISOGONAL CONJUGATE OF X₅₃)

Trilinears: \(b^{2}c^{2}\sec(B - C) : :\)

Search Value (6, 9, 13): \(-89.736436106389\)

▸ Incident Central Lines11 lines

{2, 97}, {3, 264}, {54, 69}, {76, 96}, {99, 311}, {140, 340}, {141, 287}, {160, 327}, {183, 305}, {216, 648}, {307, 320}

▸ Functional Relations12 relations

• X₉₅ = isogonal conjugate of X₅₁

• X₉₅ = isotomic conjugate of X₅

• X₉₅ = anticomplement of X₂₃₃

• X₉₅ = X₂₇₆-Ceva conjugate of X₂₇₅

• X₉₅ = cevapoint of X_i and X_j for these (i,j): (2, 3), (6, 160), (54, 97)

• X₉₅ = X_i-cross conjugate of X_j for these (i,j): (2, 276), (3, 97), (54, 275), (140, 2), (340, 1494)

• X₉₅ = intersection of tangents at X₂ and X₃ to bianticevian conic of X₂ and X₃

• X₉₅ = crosspoint of X₂ and X₃ wrt both the anticomplementary triangle and anticevian triangle of X₃

• X₉₅ = trilinear pole of line X₃₂₃X₄₀₁ (polar of X₅₃ wrt polar circle, and polar of X₆₉ wrt de Longchamps circle)

• X₉₅ = pole wrt polar circle of trilinear polar of X₅₃

• X₉₅ = X₄₈-isoconjugate (polar conjugate) of X₅₃

• X₉₅ = X₉₂-isoconjugate of X₂₁₇

Notes

Let Sym_aSym_bSym_c be the symmedial triangle. Let L_a be the reflection of line Sym_bSym_c in line BC, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₉₅. (Randy Hutson, August 19, 2015)

Let A' be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B' and C' cyclically. The triangle A'B'C' is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A', B', C' lie on the rectangular hyperbola {X₂,X₂₀,X₅₄,X₆₉,X₁₁₀,X₂₅₇₄,X₂₅₇₅,X₂₉₇₉}}. (See https://bernard-gibert.pagesperso-orange.fr/Exemples/k007.html.) Also, X₉₅ is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015)