X₁₇₇

1st MID-ARC POINT

X₁₇₇ = crosspoint of X₇ and X₁₇₄

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

X₁₇₇ (1st 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}, {9, 173}

• Related Centers: X₇ (GERGONNE POINT), X₂₃₄ (-CEVA CONJUGATE OF X₁₇₇), X₁ (INCENTER), X₁₆₇ (NAGEL POINT OF EXCENTRAL TRIANGLE), X₉ (MITTENPUNKT), X₁₇₃ (CONGRUENT ISOSCELIZERS POINT)

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

Barycentrics: \((\sin A)(\cos B/2 + \cos C/2) \sec A/2 : :\)

Search Value (6, 9, 13): \(0.975427635250\)

▸ Alternate Coordinates1 form

• Trilinears: \((b' + c')/a' : :\), where a’, b’, c’ are sidelengths of excentral triangle

▸ Incident Central Lines5 lines

{1, 167}, {7, 555}, {8, 556}, {9, 173}, {57, 164}

▸ Functional Relations8 relations

• X₁₇₇ = isogonal conjugate of X₂₆₀

• X₁₇₇ = X₇-Ceva conjugate of X₂₃₄

• X₁₇₇ = crosspoint of X₇ and X₁₇₄

• X₁₇₇ = crosssum of X₅₅ and X₂₅₉

• X₁₇₇ = SS(A→J_a)-of-X₁₀, where J_aJ_bJ_c is the excentral triangle

• X₁₇₇ = X₄-of-mid-arc triangle

• X₁₇₇ = X₁₈₂₉-of-excentral triangle

• X₁₇₇ = perspector of ABC and mid-triangle of 1st tangential mid-arc triangle and Yff central triangle

Notes

Let A',B',C' be the first points of intersection of the angle bisectors of ABC with its incircle. The tangents at A', B', C' form a triangle A″B″C″, and the lines AA″,BB″,CC″ concur in X₁₇₇. Also, X₁₇₇ = X₁ of the intouch triangle.

Clark Kimberling and G. R. Veldkamp, Problem 1160 and Solution, Crux Mathematicorum 13 (1987) 298-299 [proposed 1986].

X₁₇₇ is the perspector of ABC and the Yff central triangle, and X₁₇₇ is X₆₅-of-the-Yff-central-triangle . (Darij Grinberg, Hyacinthos #7689, 8/25/2003)