X₁₁₂₇

1st DE VILLIERS POINT

X₁₁₂₇ = Hofstadter 1/4 point

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

X₁₁₂₇ (1st DE VILLIERS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {174, 481}

• Related Centers: X₁₁₂₉ (ISOGONAL CONJUGATE OF X₁₁₂₇), X₁₇₄ (YFF CENTER OF CONGRUENCE), X₄₈₁ (1st EPPSTEIN POINT)

Trilinears: \((\sin A/4)/\sin(3A/4) : :\)

Barycentrics: 1/(cot A - cot(A/4)) : : (Nikolaos Dergiades, ADGEOM #1511, 8/19/2014)

Search Value (6, 9, 13): \(1.382356586925\)

▸ Alternate Coordinates1 form

• Trilinears: \([1 - 2 \cos(A/2)]/(1 + 2 \cos A) : :\) (M. Iliev, 5/13/07)

▸ Incident Central Lines1 line

{174, 481}

▸ Functional Relations3 relations

• X₁₁₂₇ = isogonal conjugate of X₁₁₂₉

• X₁₁₂₇ = Hofstadter 1/4 point

• X₁₁₂₇ = trilinear product of vertices of BCI triangle

Notes

Let A', B', C' be the incenters of triangles XBC, XCA, XAB, respectively, where X is the incenter, X₁. The triangle A'B'C' is perspective to ABC, and the perspector is X₁₁₂₇. Coordinates found by Darij Grinberg, 8/22/02. (The triangle A'B'C' is the BCI triangle.)

Michael de Villiers,A dual to Kosnita’s theorem, reprinted from Mathematics & Informatics Quarterly 6 (1996) 1996.