X₃₄₃

ISOTOMIC CONJUGATE OF X₂₇₅

X₃₄₃ is the isotomic conjugate of X₂₇₅.

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

X₃₄₃ (ISOTOMIC CONJUGATE OF X₂₇₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {5, 51}

• Related Centers: X₂₇₅ (CEVAPOINT OF ORTHOCENTER AND SYMMEDIAN POINT), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₅ (NINE-POINT CENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(\cot A \cos(B - C) : \cot B \cos(C - A) : \cot C \cos(A - B)\)

Barycentrics: \(\cos A \cos(B - C) : \cos B \cos(C - A) : \cos C \cos(A - B)\)

Search Value (6, 9, 13): \(-2.688803421769\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\tan B \tan C + 1 : :\)

• Barycentrics: \((b^{2} + c^{2} - a^{2}) (a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2}) : :\)

▸ Incident Central Lines13 lines

{2, 6}, {3, 68}, {5, 51}, {22, 161}, {53, 311}, {76, 297}, {140, 569}, {315, 458}, {427, 511}, {470, 634}, {471, 633}, {472, 621}, {473, 622}

▸ Functional Relations11 relations

• X₃₄₃ = isogonal conjugate of X₈₈₈₂

• X₃₄₃ = isotomic conjugate of X₂₇₅

• X₃₄₃ = complement of X₁₉₉₃

• X₃₄₃ = X_i-Ceva conjugate of X_j for these (i,j): (76, 311), (311, 5)

• X₃₄₃ = X₂₁₆-cross conjugate of X₅

• X₃₄₃ = crosspoint of X₆₉ and X₇₆

• X₃₄₃ = crosssum of X_i and X_j for these (i,j): (6, 571), (25, 32)

• X₃₄₃ = pole wrt polar circle of trilinear polar of X₈₈₈₄ (line X₄₂₁X₂₅₀₁)

• X₃₄₃ = polar conjugate of X₈₈₈₄

• X₃₄₃ = crosspoint of X₆ and X₂₉₁₇ wrt both the excentral and tangential triangles

• X₃₄₃ = X₇₅₈₀-of-orthic-triangle if ABC is acute

Notes

X₃₄₃ is the barycentric multiplier for the Johnson circumconic. (The barycentric product of X₃₄₃ and the circumcircle is the Johnson circumconic.) (Randy Hutson, August 19, 2019)