X₁₂₀

X₁₀₅-OF-MEDIAL-TRIANGLE

X₁₂₀ = X₁₀₅-of-medial triangle.

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

X₁₂₀ (X₁₀₅-OF-MEDIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 11}, {12, 85}

• Related Centers: X₄ (ORTHOCENTER), X₅₁₈ (ISOGONAL CONJUGATE OF X₁₀₅), X₂ (CENTROID), X₁₁ (FEUERBACH POINT), X₁₂ ({X), X₈₅ (ISOTOMIC CONJUGATE OF X₉)

Trilinears: \(bc[2abc - (b + c)(a^{2} + (b - c)^{2})](b^{2} + c^{2} - ab -ac) : :\)

Barycentrics: \([2abc - (b + c)(a^{2} + (b - c)^{2})](b^{2} + c^{2} - ab -ac) : :\)

Search Value (6, 9, 13): \(2.597088595260\)

X₁₂₀ lies on the nine-point circle.

▸ Incident Central Lines4 lines

{2, 11}, {10, 116}, {12, 85}, {115, 442}

▸ Functional Relations15 relations

• X₁₂₀ = complementary conjugate of X₅₁₈

• X₁₂₀ = X₄-Ceva conjugate of X₅₁₈

• X₁₂₀ = X₁₀₅-of-medial triangle.

• X₁₂₀ = complement of X₁₀₅

• X₁₂₀ = perspector of circumconic centered at X₃₂₉₀

• X₁₂₀ = center of circumconic that is locus of trilinear poles of lines passing through X₃₂₉₀

• X₁₂₀ = X₂-Ceva conjugate of X₃₂₉₀

• X₁₂₀ = polar conjugate of isogonal conjugate of X₂₀₇₂₈

• X₁₂₀ = inverse-in-excircles-radical-circle of X₃₀₃₃

• X₁₂₀ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₀₀

• X₁₂₀ = X₁₂₉₂-of-Euler-triangle

• X₁₂₀ = midpoint of X₄ and X₁₂₉₂

• X₁₂₀ = Λ(X₁, X₆), wrt orthic triangle

• X₁₂₀ = orthopole of PU(44) (line X₃X₆₆₇)

• X₁₂₀ = crosssum of circumcircle intercepts of line PU(44) (line X₃X₆₆₇)