X₂₀₆

X₂-CEVA CONJUGATE OF X₃₂

For another construction see Antreas P. Hatzipolakis and Peter Moses, Euclid 1713.

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

X₂₀₆ (X₂-CEVA CONJUGATE OF X₃₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {2, 66}

• Related Centers: X₂ (CENTROID), X₃₂ (3rd POWER POINT), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₆₆ (ISOGONAL CONJUGATE OF X₂₂)

Trilinears: \(a^{3}(b^{4} + c^{4} - a^{4}) : :\)

Search Value (6, 9, 13): \(-0.876240998547\)

▸ Alternate Coordinates1 form

• Trilinears: \((\sin A)(\sin 2A - \tan \omega ) : :\)

▸ Incident Central Lines9 lines

{2, 66}, {5, 182}, {6, 25}, {26, 511}, {69, 110}, {157, 216}, {160, 577}, {219, 692}, {237, 571}

▸ Functional Relations17 relations

• X₂₀₆ = midpoint of X_i and X_j for these (i,j): (6, 159), (110, 1177)

• X₂₀₆ = isogonal conjugate of X₁₈₀₁₈

• X₂₀₆ = isotomic conjugate of isogonal conjugate of X₂₀₉₆₈

• X₂₀₆ = complement of X₆₆

• X₂₀₆ = complementary conjugate of X₄₂₇

• X₂₀₆ = X₂-Ceva conjugate of X₃₂

• X₂₀₆ = crosspoint of X₂ and X₃₁₅

• X₂₀₆ = crosssum of X_i and X_j for these {i,j}: {2, 7391}, {6, 2353}, {66, 14376}, {339, 523}, {826, 15526}, {13854, 17407}

• X₂₀₆ = X₆₆-of-medial triangle

• X₂₀₆ = perspector of circumconic centered at X₃₂

• X₂₀₆ = isogonal conjugate of the isotomic conjugate of X₂₂

• X₂₀₆ = center of conic that is the locus of centers of conics passing through X₆ and the vertices of the tangential triangle

• X₂₀₆ = centroid of X₆ plus the vertices of the tangential triangle

• X₂₀₆ = crosssum of circumcircle intercepts of de Longchamps line

• X₂₀₆ = polar conjugate of isogonal conjugate of X₂₂₀₇₅

• X₂₀₆ = center of circumconic that is locus of trilinear poles of lines passing through X₃₂; this conic is the isogonal conjugate of the de Longchamps line

• X₂₀₆ = trilinear product X_i*X_j for these {i,j}: {2, 17453}, {6, 2172}, {9, 7251}, {19, 10316}, {22, 31}, {32, 1760}, {37, 17186}, {48, 8743}, {57, 4548}, {63, 17409}, {163, 2485}, {315, 560}, {798, 4611}, {1333, 4456}, {1397, 4123}, {1755, 11610}, {2175, 7210}, {2206, 4463}, {9247, 17907}, {9447, 17076}

Notes

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung); then X₂₀₆ = X₆-of-A'B'C'. (Randy Hutson, July 31 2018)