X₂₀₈₉

3rd MID-ARC POINT

X₂₀₈₉ = X₇-Ceva conjugate of X₁₇₄

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

X₂₀₈₉ (3rd MID-ARC POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 167}, {2, 178}

• Related Centers: X₇ (GERGONNE POINT), X₁₇₄ (YFF CENTER OF CONGRUENCE), X₁ (INCENTER), X₁₆₇ (NAGEL POINT OF EXCENTRAL TRIANGLE), X₂ (CENTROID), X₁₇₈ (2nd MID-ARC POINT)

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

Search Value (6, 9, 13): \(0.522881919355\)

▸ Alternate Coordinates2 forms

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

• Trilinears: \(\csc^{2}(A'/2) : :\), where J_aJ_bJ_c = excentral triangle

▸ Incident Central Lines3 lines

{1, 167}, {2, 178}, {7, 1488}

▸ Functional Relations7 relations

• X₂₀₈₉ = X₇-Ceva conjugate of X₁₇₄

• X₂₀₈₉ = X₁₇₃-cross conjugate of X₁₇₄

• X₂₀₈₉ = X₅₆-of-intouch triangle

• X₂₀₈₉ = X₂₄-of-mid-arc-triangle

• X₂₀₈₉ = intouch-isogonal conjugate of X₁₇₆₄₁

• X₂₀₈₉ = SS(A→J_a)-of-X₈, where J_aJ_bJ_c is the excentral triangle

• X₂₀₈₉ = trilinear pole of perspectrix of ABC and Yff central triangle

Notes

Let A', B', C' be the first points of intersection of the angle bisectors of triangle ABC with its incircle. Let A″ B″ C″ be the triangle formed by the lines tangent to the incircle at A', B', C'. Then A″B″C″ is perspective to the intouch triangle of ABC, and the perspector is X₂₀₈₉. (Darij Grinberg, Hyacinthos #8072, 10/01/03)