X₂₅₈

CONGRUENT INCIRCLES ISOSCELIZER POINT

In Yff’s isoscelizer configuration, if X = X₂₅₈, then the isosceles triangles T_A, T_B, T_C have congruent incircles.

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

X₂₅₈ (CONGRUENT INCIRCLES ISOSCELIZER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 164}, {57, 173}

• Related Centers: X₂₅₉ (ISOGONAL CONJUGATE OF X₁₇₄), X₁ (INCENTER), X₁₆₄ (INCENTER OF EXCENTRAL TRIANGLE), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₁₇₃ (CONGRUENT ISOSCELIZERS POINT)

Trilinears: \(1/(\cos B/2 + \cos C/2 - \cos A/2) : :\)

Search Value (6, 9, 13): \(2.578860774943\)

▸ Alternate Coordinates7 forms

• Trilinears: \(1 + \sin(B/2) + \sin(C/2) - \sin(A/2) : :\)

• Trilinears: \(\tan(A/2) - \sec(A/2) : :\)

• Trilinears: \(\tan(B/2) - \sec(B/2) : \tan(C/2) - \sec(C/2) : :\)

• Trilinears: \(1/(b' + c' - a') : :\), where J_aJ_bJ_c is the excentral triangle

• Trilinears: \(\tan A'/2 : :\), where J_aJ_bJ_c is the excentral triangle

• Trilinears: \(\cot A' - \csc A' : :\), where J_aJ_bJ_c is the excentral triangle

• Trilinears: (distance from A to A-excircle) : :

▸ Incident Central Lines3 lines

{1, 164}, {57, 173}, {259, 289}

▸ Functional Relations9 relations

• X₂₅₈ = isogonal conjugate of X₁₇₃

• X₂₅₈ = X₂₅₉-cross conjugate of X₁

• X₂₅₈ = X₃₆₆-aleph conjugate of X₃₆₃

• X₂₅₈ = SS(a→a’) of X₅₇, where J_aJ_bJ_c is the excentral triangle (trilinear substitution)

• X₂₅₈ = X₃₃-of-excentral-triangle

• X₂₅₈ = homothetic center of ABC and intangents triangle of excentral triangle

• X₂₅₈ = insimilicenter of incircle and incircle of excentral triangle

• X₂₅₈ = {X₁,X₁₆₄}-harmonic conjugate of X₈₀₇₈

• X₂₅₈ = perspector of ABC and the extouch triangle of the intouch triangle