X₁₂₂

X₁₀₇-OF-MEDIAL-TRIANGLE

X₁₂₂ = reflection of X₁₃₃ in X₅

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

X₁₂₂ (X₁₀₇-OF-MEDIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {133, 5}

• Related Centers: X₁₃₃ (INTERSECTION OF LINES X), X₅ (NINE-POINT CENTER)

Trilinears: \((b^{2} - c^{2})^{2}(\cos A - \cos B \cos C) \cot^{2} A : :\)

Barycentrics: \(a(b^{2} - c^{2})^{2}(\cos A - \cos B \cos C) \cot^{2} A : :\)

Search Value (6, 9, 13): \(3.821374194765\)

▸ Alternate Coordinates1 form

• Barycentrics: \((b^{2} - c^{2})^{2} (a^{2} - b^{2} - c^{2})^{2} (3 a^{4} - 2 a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2}) : :\)

X₁₂₂ lies on the nine-point circle, the cevian circle of X₂₀.

▸ Incident Central Lines6 lines

{2, 107}, {3, 113}, {5, 133}, {118, 440}, {125, 684}, {138, 233}

▸ Functional Relations20 relations

• X₁₂₂ = reflection of X₁₃₃ in X₅

• X₁₂₂ = isogonal conjugate of X₁₅₃₈₄

• X₁₂₂ = complementary conjugate of X₅₂₀

• X₁₂₂ = X_i-Ceva conjugate of X_j for these (i,j): (4, 520), (253, 525)

• X₁₂₂ = crosssum of X_i and X_j for these (i,j): (64, 1301), (112, 154)

• X₁₂₂ = crosspoint of X₂₅₃ and X₅₂₅

• X₁₂₂ = crossdifference of every pair of points on line X₁₁₂X₁₃₀₁

• X₁₂₂ = X₁₀₇-of-medial triangle

• X₁₂₂ = center of the rectangular hyperbola that passes through A, B, C, and X₂₀

• X₁₂₂ = X₁₂₉₃-of-orthic-triangle if ABC is acute

• X₁₂₂ = complement of X₁₀₇

• X₁₂₂ = perspector of orthic triangle and tangential triangle of hyperbola {A, B, C,X₂,X₃}}

• X₁₂₂ = inverse-in-polar-circle of X₁₃₀₁

• X₁₂₂ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₁₂₉₇

• X₁₂₂ = inverse-in-Moses-radical-circle of X₃₃₅₀₄

• X₁₂₂ = X₂-Ceva conjugate of X₆₅₈₇

• X₁₂₂ = barycentric product X₂₀*X₁₅₅₂₆ = X₂₀*X₅₂₅^2

• X₁₂₂ = crosssum of circumcircle intercepts of line X₃X₆₄

• X₁₂₂ = orthopole of line X₃X₆₄

• X₁₂₂ = Kirikami-six-circles image of X₂₀