X₆₁₅

ISOGONAL CONJUGATE OF X₅₈₉

X₆₁₅ is the isogonal conjugate of X₅₈₉.

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

X₆₁₅ (ISOGONAL CONJUGATE OF X₅₈₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 491}

• Related Centers: X₂ (CENTROID), X₄₉₁ (CEVAPOINT OF X₂ AND X₄₈₇)

Trilinears: \((a^{2} - 4 \operatorname{area})/a : :\)

Barycentrics: \(a^{2} - 4 \operatorname{area} : :\)

Search Value (6, 9, 13): \(-228.134102794607\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sin A - 2 \sin B \sin : :\)

• Trilinears: \(aR- bc : :\)

• Barycentrics: \(a^{2} - 2S : :\)

▸ Incident Central Lines10 lines

{1, 3300}, {2, 6}, {3, 486}, {4, 1152}, {5, 372}, {32, 639}, {39, 641}, {140, 371}, {605, 750}, {606, 748}

▸ Functional Relations9 relations

• X₆₁₅ = isogonal conjugate of X₅₈₉

• X₆₁₅ = complement of X₄₉₁

• X₆₁₅ = crosspoint of X₂ and X₄₈₆

• X₆₁₅ = crosssum of X₆ and X₃₇₂

• X₆₁₅ = perspector of circumconic centered at X₆₄₂

• X₆₁₅ = center of circumconic that is locus of trilinear poles of lines passing through X₆₄₂

• X₆₁₅ = X₂-Ceva conjugate of X₆₄₂

• X₆₁₅ = homothetic center of inner Vecten and 4th tri-squares triangles

• X₆₁₅ = homothetic center of [outer Vecten of inner Vecten triangle] and pedal triangle of X₁₁₅₂