X₁₄₇₉

{X₁,X₄}-HARMONIC CONJUGATE OF X₁₄₇₈

X₁₄₇₉ is the {X₁,X₄}-harmonic conjugate of X₁₄₇₈.

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

X₁₄₇₉ ({X₁,X₄}-HARMONIC CONJUGATE OF X₁₄₇₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {3, 11}

• Related Centers: X₁₀₆₇ (ISOGONAL CONJUGATE OF X₁₀₆₆), X₁ (INCENTER), X₄ (ORTHOCENTER), X₃ (CIRCUMCENTER), X₁₁ (FEUERBACH POINT)

Trilinears: \(1 - 2 \cos B \cos C : :\)

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

Search Value (6, 9, 13): \(3.865047378943\)

▸ Linear Combinations2 combinations

• X₁₄₇₉ = \((R/r)\) X₁ - \(3\) X₂ + \(2\) X₃

• X₁₄₇₉ = \(R\) X₁ + \(r\) X₃ - \(2r\) X₅

▸ Incident Central Lines22 lines

{1, 4}, {2, 35}, {3, 11}, {5, 55}, {7, 79}, {8, 80}, {12, 381}, {20, 36}, {30, 56}, {46, 516}, {63, 90}, {148, 330}, {156, 215}, {315, 350}, {377, 1125}, {382, 999}, {387, 1203}, {442, 1001}, {495, 546}, {528, 1329}, {614, 1370}, {1387, 1388}

▸ Functional Relations7 relations

• X₁₄₇₉ = reflection of X_i in X_j for these (i,j): (46, 1210), (56, 496)

• X₁₄₇₉ = X₁₀₆₇-Ceva conjugate of X₁

• X₁₄₇₉ = homothetic center of intangents triangle and reflection of tangential triangle in X₅

• X₁₄₇₉ = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

• X₁₄₇₉ = homothetic center of Ehrmann mid-triangle and 1st Johnson-Yff triangle

• X₁₄₇₉ = Ursa-major-to-Ursa-minor similarity image of X₃

• X₁₄₇₉ = homothetic center of 2nd isogonal triangle of X₁ and the reflection of the anticomplementary triangle in X₄; see X₃₆

Notes

Peter Yff, “Three concurrent congruent circles ‘inscribed’ in a triangle,” manuscript, 1998; X₁₄₇₉ is the point C' on page 5. See also X₄₉₅-X₄₉₉.