X₁₅₉₇

POINT CHERTAN I

X₁₅₉₇ = orthocentroidal-circle-inverse of X₁₅₉₆

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

X₁₅₉₇ (POINT CHERTAN I) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {64, 389}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₃₈₉ (CENTER OF THE TAYLOR CIRCLE)

Trilinears: \(\cos A + 2 \sec A : :\)

Search Value (6, 9, 13): \(-12.385457772305\)

▸ Incident Central Lines4 lines

{2, 3}, {33, 999}, {64, 389}, {578, 1498}

▸ Functional Relations6 relations

• X₁₅₉₇ = complement of X₃₅₅₁₃

• X₁₅₉₇ = circumcircle-inverse of X₃₇₉₃₄

• X₁₅₉₇ = orthocentroidal-circle-inverse of X₁₅₉₆

• X₁₅₉₇ = {X₃,X₄}-harmonic conjugate of X₁₅₉₈

• X₁₅₉₇ = polar conjugate of isogonal conjugate of X₃₃₈₇₁

• X₁₅₉₇ = center of conic that is locus of {X₄,P}-harmonic conjugate of Q, where P and Q lie on the circumcircle and are collinear with X₄

Notes

As a point on the Euler line, X₁₅₉₇ has Shinagawa coefficients \((F, 2E - F)\).