X₃₇₈

REFLECTION OF X₂₂ IN X₃

X₃₇₈ is the reflection of X₂₂ in X₃.

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

X₃₇₈ (REFLECTION OF X₂₂ IN X₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {22, 3}, {2, 3} (Euler line)

• Related Centers: X₂₂ (EXETER POINT), X₃ (CIRCUMCENTER), X₂ (CENTROID)

Trilinears: \(\sec A + 2 \cos A : \sec B + 2 \cos B : \sec C + 2 \cos C\)

Barycentrics: \(\tan A + \sin 2A : \tan B + \sin 2B : \tan C + \sin 2C : :\)

Search Value (6, 9, 13): \(37.928175425362\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₃₇₈ = \(6\) X₂ + \((J^{2} + 3)\) X₃ = \(2\) X₃ - X₂₂

▸ Incident Central Lines11 lines

{1, 1063}, {2, 3}, {6, 74}, {33, 36}, {34, 35}, {54, 64}, {99, 264}, {185, 578}, {232, 574}, {477, 935}, {847, 1105}

▸ Functional Relations7 relations

• X₃₇₈ = reflection of X_i in X_j for these (i,j): (4, 427), (22, 3)

• X₃₇₈ = isogonal conjugate of X₄₈₄₆

• X₃₇₈ = anticomplement of X₁₅₇₆₀

• X₃₇₈ = inverse-in-orthocentroidal-circle of X₄₀₃

• X₃₇₈ = harmonic center of circumcircle and polar circle

• X₃₇₈ = polar conjugate of X₃₄₂₈₉

• X₃₇₈ = endo-homothetic center of X₄-anti-altimedial and anti-orthocentroidal triangles

Notes

As a point on the Euler line, X₃₇₈ has Shinagawa coefficients \((2F, E - 2F)\).

X₃₇₈ is the {X₃,X₄}-harmonic conjugate of X₂₄.