X₃₈₂

REFLECTION OF CIRCUMCENTER IN ORTHOCENTER

X₃₈₂ is the reflection of circumcenter in orthocenter.

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

X₃₈₂ (REFLECTION OF CIRCUMCENTER IN ORTHOCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 4} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER)

Trilinears: \(\cos A - 4 \cos B \cos C : \cos B - 4 \cos C \cos A : \cos C - 4 \cos A \cos B\)

Barycentrics: \(a(\cos A - 4 \cos B \cos C) : b(\cos B - 4 \cos C \cos A) : c(\cos C - 4 \cos A \cos B)\)

Search Value (6, 9, 13): \(-18.135601716049\)

▸ Alternate Coordinates7 forms

• Trilinears: \(5 \cos A - 4 \sin B \sin C : 5 \cos B - 4 \sin C \sin A : 5 \cos C - 4 \sin A \sin B\)

• Trilinears: \(4 \sec A - \sec B \sec C : :\)

• Trilinears: \(4 \cos A - \cos B \cos C - 3 \sin B \sin C : :\)

• Trilinears: \(\cos A + 6 \cos B \cos C - 2 \sin B \sin C : :\)

• Trilinears: \(4 \cos B \cos C - 2 \sin B \sin C + \cos(B - C) : :\)

• Trilinears: \(5 \cos A - \cos(B - C) + \cos B \cos C - 3 \sin B \sin C : :\)

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

▸ Linear Combinations1 combination

• X₃₈₂ = \(6\) X₂ - \(5\) X₃ = X₃ - \(2\) X₄

▸ Incident Central Lines7 lines

{2, 3}, {64, 265}, {155, 399}, {185, 568}, {195, 1498}, {355, 516}, {952, 962}

▸ Functional Relations14 relations

• X₃₈₂ = reflection of X_i in X_j for these (i,j): (3, 4), (20, 5), (110, 1539), (550, 546), (3534, 381)

• X₃₈₂ = inverse-in-orthocentroidal-circle of X₅₄₆

• X₃₈₂ = complement of X₃₅₂₉

• X₃₈₂ = anticomplement of X₅₅₀

• X₃₈₂ = Kosnita-to-tangential similarity image of X₄

• X₃₈₂ = homothetic center of Johnson triangle and mid-triangle of medial and anticomplementary triangles

• X₃₈₂ = {X₂,X₃}-harmonic conjugate of X₁₅₇₂₀

• X₃₈₂ = {X₃,X₄}-harmonic conjugate of X₃₈₁

• X₃₈₂ = Ehrmann-mid-to-ABC similarity image of X₃

• X₃₈₂ = homothetic center of Ehrmann vertex-triangle and anti-Hutson intouch triangle

• X₃₈₂ = homothetic center of Ehrmann side-triangle and anti-excenters-incenter reflections triangle

• X₃₈₂ = homothetic center of Ehrmann mid-triangle and ABC-X3 reflections triangle

• X₃₈₂ = {X₄,X₂₄}-harmonic conjugate of X₃₇₁₉₇

• X₃₈₂ = {X₂,X₃₅₂₉}-harmonic conjugate of X₅₅₀

Notes

As a point on the Euler line, X₃₈₂ has Shinagawa coefficients \((1,-5)\).

Let Ref_aRef_bRef_c be the reflection triangle. Let A″ be the trilinear pole, wrt Ref_aRef_bRef_c, of line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₈₂. (Randy Hutson, January 29, 2015)

X₃₈₂ is the {X₅,X₂₀}-harmonic conjugate of X₃.