X₄₉₁

CEVAPOINT OF X₂ AND X₄₈₇

X₄₉₁ is the cevapoint of X₂ and X₄₈₇.

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

X₄₉₁ (CEVAPOINT OF X₂ AND X₄₈₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 615}

• Related Centers: X₂ (CENTROID), X₆₁₅ (ISOGONAL CONJUGATE OF X₅₈₉)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\cos A)(1 - \csc A \sin B \sin C) + \cos B \cos C\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(464.156311966686\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\sin(A - \pi /4)\csc^{2} A : \sin(B - \pi /4)\csc^{2} B : \sin(C - \pi /4)\csc^{2} C\) (M. Iliev, 4/12/2007)

• Trilinears: \((1 - \cot A) \csc A : (1 - \cot B) \csc B : (1 - \cot C) \csc C\) (M. Iliev, 4/12/2007)

• Trilinears: \((S - S_{A})/a : (S - S_{B})/b : (S - S_{C})/c\) (C. Lozada, 8/07/2013)

• Barycentrics: \(b^{2} + c^{2} - a^{2} - 4\sigma : c^{2} + a^{2} - b^{2} - 4\sigma : a^{2} + b^{2} - c^{2} - 4\sigma\) (M. Iliev, 5/13/07)

▸ Incident Central Lines8 lines

{2, 6}, {3, 490}, {4, 487}, {5, 637}, {76, 485}, {315, 371}, {372, 642}, {488, 631}

▸ Functional Relations7 relations

• X₄₉₁ = isotomic conjugate of X₄₈₆

• X₄₉₁ = anticomplement of X₆₁₅

• X₄₉₁ = X₂₆₄-Ceva conjugate of X₄₉₂

• X₄₉₁ = cevapoint of X₂ and X₄₈₇

• X₄₉₁ = crosspoint of X₂ and X₄₈₇ wrt both the excentral and anticomplementary triangles

• X₄₉₁ = {X₂,X₆₉}-harmonic conjugate of X₄₉₂

• X₄₉₁ = homothetic center of ABC and unary cofactor triangle of 2nd Kenmotu diagonals triangle

Notes

X₄₉₁ is a pole associated with squares that circumscribe ABC. For details and reference, see X₄₈₈. (Floor van Lamoen, 4/26/98)