X₃₉₀

REFLECTION OF GERGONNE POINT IN INCENTER

X₃₉₀ is the reflection of Gergonne point in incenter.

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

X₃₉₀ (REFLECTION OF GERGONNE POINT IN INCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 1}

• Related Centers: X₇ (GERGONNE POINT), X₁ (INCENTER)

Trilinears: \(bc(b + c - a)[3a^{2} + (b - c)^{2}]\)

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

Search Value (6, 9, 13): \(2.636456039908\)

▸ Linear Combinations1 combination

• X₃₉₀ = \(4(R/r)\) X₁ - \(3\) X₂ + \(4\) X₃

▸ Incident Central Lines13 lines

{1, 7}, {2, 11}, {3, 1058}, {4, 495}, {8, 9}, {30, 1056}, {40, 938}, {144, 145}, {376, 999}, {387, 595}, {496, 631}, {944, 971}, {952, 1000}

▸ Functional Relations8 relations

• X₃₉₀ = midpoint of X₁₄₄ and X₁₄₅

• X₃₉₀ = reflection of X_i in X_j for these (i,j): (7, 1), (8, 9)

• X₃₉₀ = anticomplement of X₂₅₅₀

• X₃₉₀ = crossdifference of every pair of points on line X₆₅₇X₆₆₅

• X₃₉₀ = inverse-in-Feuerbach-hyperbola of X₂

• X₃₉₀ = {X₁,X₂₀}-harmonic conjugate of X₃₆₀₀

• X₃₉₀ = {X₁₇₅,X₁₇₆}-harmonic conjugate of X₂₇₉

• X₃₉₀ = bicentric sum of PU(122)

Notes

X₃₉₀ is the point in whIch the extended legs X₁X₇ and X₈X₉ of the trapezoid X₁X₇X₈X₉ meet. (Randy Hutson, February 10, 2016)

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 anticomplementary triangle at X₃₉₀. (Randy Hutson, June 7, 2019)