X₁₆₂

CEVAPOINT OF X₁₀₈ AND X₁₀₉

X₁₆₂ is the cevapoint of X₁₀₈ and X₁₀₉.

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

X₁₆₂ (CEVAPOINT OF X₁₀₈ AND X₁₀₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {29, 58}, {28, 88}

• Related Centers: X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER)), X₁₀₉ (Ψ(INCENTER, CIRCUMCENTER)), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₅₈ (ISOGONAL CONJUGATE OF X₁₀), X₂₈ (CEVAPOINT OF X₁₉ AND X₂₅), X₈₈ (ISOGONAL CONJUGATE OF X₄₄)

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

Barycentrics: \(a/(\sin 2B - \sin 2C) : b/(\sin 2C - \sin 2A) : c/(\sin 2A - \sin 2B)\)

Search Value (6, 9, 13): \(-0.434609584744\)

▸ Alternate Coordinates3 forms

• Trilinears: \(1/[(b^{2} - c^{2})(b^{2} + c^{2} - a^{2})]\)

• Trilinears: \(\sec A \csc(B - C) : :\)

• Trilinears: \(1/(\tan B - \tan C) : :\)

▸ Incident Central Lines19 lines

{4, 270}, {6, 1013}, {19, 897}, {27, 673}, {28, 88}, {29, 58}, {31, 92}, {47, 158}, {63, 204}, {100, 112}, {107, 109}, {108, 110}, {190, 643}, {238, 415}, {240, 896}, {242, 422}, {255, 1099}, {412, 580}, {799, 811}

▸ Functional Relations20 relations

• X₁₆₂ = isogonal conjugate of X₆₅₆

• X₁₆₂ = isotomic conjugate of X₁₄₂₀₈

• X₁₆₂ = complement of X₃₄₈₄₆

• X₁₆₂ = X₂-Ceva conjugate of X₃₉₀₅₂

• X₁₆₂ = X₂₅₀-Ceva conjugate of X₂₇₀

• X₁₆₂ = cevapoint of X_i and X_j for this (i,j): (108,109)

• X₁₆₂ = X_i-cross conjugate of X_j for these (i,j): (108, 107), (109, 110)

• X₁₆₂ = crosssum of X₈₁₀ and X₈₂₂

• X₁₆₂ = X_i-aleph conjugate of X_j for these (i,j): (28, 1052), (107, 920), (162, 1), (648, 63)

• X₁₆₂ = trilinear pole of line X₁X₁₉

• X₁₆₂ = trilinear product of X₁₁₁₃ and X₁₁₁₄

• X₁₆₂ = crossdifference of PU(75)

• X₁₆₂ = pole wrt polar circle of trilinear polar of X₁₅₇₇ (line X₁₁₀₉X₂₆₃₂)

• X₁₆₂ = X₄₈-isoconjugate (polar conjugate) of X₁₅₇₇

• X₁₆₂ = X₉₂-isoconjugate of X₈₂₂

• X₁₆₂ = X₆-isoconjugate of X₅₂₅

• X₁₆₂ = crosspoint of X₈₁₁ and X₈₂₃

• X₁₆₂ = trilinear product of PU(74)

• X₁₆₂ = perspector of conic {A, B, C,PU(74)}}

• X₁₆₂ = barycentric product of circumcircle intercepts of line X₁₉X₂₇

Notes

Let L_a be the A-extraversion of line X₁X₁₉, and define L_b and L_c cyclically. Let A' = L_b ∩ L_c, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₆₂. (Randy Hutson, January 29, 2018)

Let L_a be the A-extraversion of line X₈X₂₉, and define L_b and L_c cyclically. Let A' = L_b ∩ L_c, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₆₂. (Randy Hutson, January 29, 2018)

Let L_a be the A-extraversion of line X₉X₂₁, and define L_b and L_c cyclically. Let A' = L_b ∩ L_c, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₆₂. (Randy Hutson, January 29, 2018)

See (and hear) Dan Reznik’s Dance of the Swans: X₈₈ and X₁₆₂ and their Never-Touching Motion Over the Elliptic Billiard (March 4, 2020)