X₁₅₄

X₃-CEVA CONJUGATE OF X₆

X₁₅₄ is the {X₂₆,X₁₅₆}-harmonic conjugate of X₁₅₅. For a list of harmonic conjugates of X₁₅₄, click Tables at the top of this page.

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

X₁₅₄ (X₃-CEVA CONJUGATE OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 25}, {3, 64}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₆₄ (ISOGONAL CONJUGATE OF X₂₀)

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

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

Search Value (6, 9, 13): \(-1.499290829751\)

▸ Alternate Coordinates3 forms

• Trilinears: \(a(\tan B + \tan C - \tan A) : b(\tan C + \tan A - \tan B) : c(\tan A + \tan B - \tan C)\)

• Trilinears: \((\sec A - \sec B \sec C)a^{2} : :\)

• Barycentrics: \((\sin^{2} A)(\tan B + \tan C - \tan A) : :\)

▸ Incident Central Lines11 lines

{3, 64}, {6, 25}, {22, 110}, {26, 155}, {31, 56}, {48, 55}, {160, 418}, {197, 692}, {198, 212}, {205, 220}, {237, 682}

▸ Functional Relations10 relations

• X₁₅₄ = isogonal conjugate of X₂₅₃

• X₁₅₄ = complement of X₃₂₀₆₄

• X₁₅₄ = X₃-Ceva conjugate of X₆

• X₁₅₄ = crosssum of X_i and X_j for these (i,j): (64, 1073), (122, 525)

• X₁₅₄ = X₁₀₉-beth conjugate of X₁₅₄

• X₁₅₄ = X₂-of-tangential triangle

• X₁₅₄ = centroid of cevian triangle of X₂₀

• X₁₅₄ = pole wrt circumcircle of the trilinear polar of X₃

• X₁₅₄ = Thomson isogonal conjugate of X₄

• X₁₅₄ = intouch-to-ABC functional image of X₂

Notes

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the Ara triangle at X₁₅₄.