X₁₃₇₄

2nd GRIFFITHS POINT

X₁₃₇₄ = perspector of outer Soddy triangle and cross-triangle of inner and outer Soddy triangles

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

X₁₃₇₄ (2nd GRIFFITHS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 7}, {226, 3316}

• Related Centers: X₁ (INCENTER), X₇ (GERGONNE POINT), X₂₂₆ (-CEVA CONJUGATE OF X₆₅), X₃₃₁₆ (ISOGONAL CONJUGATE OF X₃₃₁₁)

Trilinears: \(1 - 8(\operatorname{area} ABC)/[a(b + c - a)] : :\)

Search Value (6, 9, 13): \(0.472226176610\)

▸ Alternate Coordinates1 form

• Trilinears: \(1 - 4 \sec A/2 \cos B/2 \cos C/2 : :\)

▸ Linear Combinations1 combination

• X₁₃₇₄ = \(2s\) X₁ - \((2r + 8R)\) X₇

▸ Incident Central Lines2 lines

{1, 7}, {226, 3316}

▸ Functional Relations6 relations

• X₁₃₇₄ = {X₁,X₇}-harmonic conjugate of X₁₃₇₃

• X₁₃₇₄ = {X₇,X₁₇₅}-harmonic conjugate of X₄₈₂

• X₁₃₇₄ = {X₇,X₄₈₁}-harmonic conjugate of X₁

• X₁₃₇₄ = {X₄₈₁,X₄₈₂}-harmonic conjugate of X₁₇₅

• X₁₃₇₄ = perspector of outer Soddy triangle and cross-triangle of inner and outer Soddy triangles

• X₁₃₇₄ = perspector of outer Soddy tangential triangle and cross-triangle of inner and outer Soddy tangential triangles

Notes

Adrian Oldknow, “The Euler-Gergonne-Soddy Triangle of a Triangle,” American Mathematical Monthly 103 (1996) 319-329. See page 327.