X₁₃₇₃

1st GRIFFITHS POINT

X₁₃₇₃ = perspector of inner Soddy triangle and cross-triangle of inner and outer Soddy triangles

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

X₁₃₇₃ (1st 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, 3317}

• Related Centers: X₁ (INCENTER), X₇ (GERGONNE POINT), X₂₂₆ (-CEVA CONJUGATE OF X₆₅), X₃₃₁₇ (ISOGONAL CONJUGATE OF X₃₃₁₂)

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

Search Value (6, 9, 13): \(0.916837142015\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₁₃₇₃ = \(2s\) X₁ + \((2r + 8R)\) X₇

▸ Incident Central Lines2 lines

{1, 7}, {226, 3317}

▸ Functional Relations3 relations

• X₁₃₇₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (1, 7, 1374), (7, 176, 481), (7, 482, 1), (175, 482, 1), (176, 481, 1), (481, 482, 176)

• X₁₃₇₃ = perspector of inner Soddy triangle and cross-triangle of inner and outer Soddy triangles

• X₁₃₇₃ = perspector of inner 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.

See also MathWorld, Griffiths Points.