X₃₄₈

ISOTOMIC CONJUGATE OF X₂₈₁

X₃₄₈ is the isotomic conjugate of X₂₈₁.

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

X₃₄₈ (ISOTOMIC CONJUGATE OF X₂₈₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 21}, {2, 85}

• Related Centers: X₂₈₁ (-CROSS CONJUGATE OF X₉), X₇ (GERGONNE POINT), X₂₁ (SCHIFFLER POINT), X₂ (CENTROID), X₈₅ (ISOTOMIC CONJUGATE OF X₉)

Trilinears: \(\cot A \sec^{2}(A/2) : :\)

Barycentrics: \(1/(1 + \sec A) : 1/(1 + \sec B) : 1/(1 + \sec C)\)

Search Value (6, 9, 13): \(-468.904101327182\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sec^{2}(C/2)\)

• Trilinears: \((\csc A)/(1 + \sec A) : :\)

• Barycentrics: \((b^{2} + c^{2} - a^{2})/(b + c - a) : :\)

▸ Incident Central Lines10 lines

{2, 85}, {7, 21}, {8, 664}, {69, 73}, {75, 280}, {150, 944}, {201, 337}, {274, 278}, {304, 345}, {499, 1111}

▸ Functional Relations9 relations

• X₃₄₈ = isogonal conjugate of X₆₀₇

• X₃₄₈ = isotomic conjugate of X₂₈₁

• X₃₄₈ = complement of X₃₀₆₉₄

• X₃₄₈ = anticomplement of polar conjugate of X₃₄₃₉₈

• X₃₄₈ = X₂₇₄-Ceva conjugate of X₈₅

• X₃₄₈ = cevapoint of X_i and X_j for these (i,j): (2, 347), (63, 77)

• X₃₄₈ = X₂₂₂-cross conjugate of X₇

• X₃₄₈ = polar conjugate of X₁₈₅₇

• X₃₄₈ = trilinear pole of line X₄₀₂₅X_4131

Notes

X₃₄₈ is the Brianchon point (perspector) of the inconic that is the isotomic conjugate of the polar conjugate of the incircle. The center of the inconic is X₁₇₀₇₃. (Randy Hutson, November 30, 2018)

Let A₃₆B₃₆C₃₆ be Gemini triangle 36. Let A' be the perspector of conic {A, B, C,B₃₆,C₃₆}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₄₈. (Randy Hutson, January 15, 2019)