X₄₉₇

CROSSPOINT OF GERGONNE POINT AND NAGEL POINT

X₄₉₇ is the crosspoint of Gergonne point and Nagel point.

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

X₄₉₇ (CROSSPOINT OF GERGONNE POINT AND NAGEL POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {2, 11}

• Related Centers: X₇ (GERGONNE POINT), X₈ (NAGEL POINT), X₁ (INCENTER), X₄ (ORTHOCENTER), X₂ (CENTROID), X₁₁ (FEUERBACH POINT)

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

Barycentrics: \((\sin A)(1 - \cos B \cos C) : (\sin B)(1 - \cos C \cos A) : (\sin C)(1 - \cos A \cos B)\)

Search Value (6, 9, 13): \(2.637823354522\)

▸ Alternate Coordinates2 forms

• Barycentrics: \((S_{\omega} - bc)(s - a) : :\)

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

▸ Linear Combinations1 combination

• X₄₉₇ = \(2(R/r)\) X₁ - \(3\) X₂ + \(2\) X₃

▸ Incident Central Lines20 lines

{1, 4}, {2, 11}, {3, 496}, {7, 354}, {8, 210}, {20, 56}, {29, 1036}, {30, 999}, {35, 499}, {36, 376}, {57, 516}, {65, 938}, {69, 350}, {80, 1000}, {212, 238}, {329, 518}, {381, 495}, {452, 958}, {614, 1040}, {1057, 1065}

▸ Functional Relations11 relations

• X₄₉₇ = isogonal conjugate of X₁₀₃₇

• X₄₉₇ = isotomic conjugate of X₈₈₁₇

• X₄₉₇ = anticomplement of X₁₃₇₆

• X₄₉₇ = crosspoint of X_i and X_j for these (i,j): (7, 8), (29, 314)

• X₄₉₇ = crosssum of X_i and X_j for these (i,j): (55, 56), (73, 1402)

• X₄₉₇ = crossdifference of every pair of points on line X₆₅₂X₆₆₅

• X₄₉₇ = homothetic center of Mandart-incircle triangle and anticomplementary triangle

• X₄₉₇ = inverse-in-Feuerbach-hyperbola of X₂₅₅₀

• X₄₉₇ = homothetic center of anticomplementary triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

• X₄₉₇ = homothetic center of 2nd Johnson-Yff triangle triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

• X₄₉₇ = {X₁,X₄}-harmonic conjugate of X₃₈₈

Notes

Let A'B'C' be the extangents-to-intangents similarity image of ABC. A'B'C' is homothetic to ABC at X₅₅ and to the medial triangle at X₄₉₇. (Randy Hutson, March 21, 2019)