X₄₁

X₆-CEVA CONJUGATE OF X₃₁

X₄₁ = trilinear product of vertices of 4th mixtilinear triangle

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

X₄₁ (X₆-CEVA CONJUGATE OF X₃₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {9, 21}, {31, 32}

• Related Centers: X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT), X₉ (MITTENPUNKT), X₂₁ (SCHIFFLER POINT), X₃₂ (3rd POWER POINT)

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

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

Search Value (6, 9, 13): \(1.800613978898\)

▸ Alternate Coordinates3 forms

• Trilinears: \(a^{2}\cot(A/2) : b^{2}\cot(B/2) : c^{2}\cot(C/2)\)

• Trilinears: \(a^{2}(a - s) : b^{2}(b - s) : c^{2}(c - s)\)

• Trilinears: \(a \tan A' : :\), where J_aJ_bJ_c is the excentral triangle

▸ Linear Combinations1 combination

• X₄₁ = \((r + 2R)(r^{2} + 4rR + s^{2})\) X₁ - \(6rR(r + 4R)\) X₂ - \(2r(^{2} + 4rR - s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines16 lines

{1, 101}, {3, 218}, {6, 48}, {9, 21}, {25, 42}, {31, 32}, {37, 584}, {55, 220}, {58, 609}, {65, 910}, {219, 1036}, {226, 379}, {560, 872}, {601, 906}, {603, 911}, {663, 884}

▸ Functional Relations18 relations

• X₄₁ = isogonal conjugate of X₈₅

• X₄₁ = isotomic conjugate of X₂₀₅₆₇

• X₄₁ = complement of X₂₁₂₈₅

• X₄₁ = anticomplement of X₁₇₀₄₆

• X₄₁ = X_i-Ceva conjugate of X_j for these (i,j): (6, 31), (9, 212), (284, 55)

• X₄₁ = crosspoint of X_i and X_j for these (i,j): (6, 55), (9, 33)

• X₄₁ = crosssum of X_i and X_j for these (i,j): (1, 169), (2, 7), (57, 77), (92, 342), (226, 1441), (514, 1111)

• X₄₁ = crossdifference of every pair of points on line X₅₂₂X₆₉₃

• X₄₁ = X_i-beth conjugate of X_j for these (i,j): (41, 32), (101, 41), (220, 220)

• X₄₁ = X₇₅-isoconjugate of X₅₇

• X₄₁ = X₉₂-isoconjugate of X₇₇

• X₄₁ = trilinear product of vertices of 4th mixtilinear triangle

• X₄₁ = trilinear product of vertices of 5th mixtilinear triangle

• X₄₁ = trilinear product of PU(93)

• X₄₁ = barycentric product of PU(104)

• X₄₁ = PU(93)-harmonic conjugate of X₆₆₃

• X₄₁ = bicentric sum of PU(93)

• X₄₁ = perspector of unary cofactor triangles of Gemini triangles 1 and 13

Notes

For an artistic design generated by X₄₁, see X₂₄₄.

X₄₁ is the {X₃₂,X₂₁₃}-harmonic conjugate of X₃₁.