X₁₈₆

INVERSE-IN-CIRCUMCIRCLE OF X₄

X₁₈₆ = circumcircle-inverse of X₄

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

X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Central Lines: {3, 4} (Euler line)

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

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

Barycentrics: \((\sin A)(4 \cos A - \sec A) : (\sin B)(4 \cos B - \sec B) : (\sin C)(4 \cos C - \sec C)\)

Tripolars: \(|\cos A| : :\)

Search Value (6, 9, 13): \(3.503719756839\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sin 3A \csc 2A : \sin 3B \csc 2B : \sin 3C \csc 2C\)

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

▸ Linear Combinations1 combination

• X₁₈₆ = \(3\) X₂ + \((J^{2} - 3)\) X₃ = \(2\) X₃ + X₂₃ = \((a^{2} - b^{2} - c^{2})(a^{2} + b^{2} - c^{2})(a^{2} - b^{2} + c^{2})\) X₃ + \((a^{2}b^{2}c^{2})\) X₄

▸ Incident Central Lines8 lines

{2, 3}, {54, 389}, {93, 252}, {98, 935}, {107, 477}, {112, 187}, {249, 250}, {2931, 3580}

▸ Functional Relations31 relations

• X₁₈₆ = reflection of X_i in X_j for these (i,j): (4, 403), (403, 468)

• X₁₈₆ = isogonal conjugate of X₂₆₅

• X₁₈₆ = isotomic conjugate of X₃₂₈

• X₁₈₆ = complement of X₃₁₅₃

• X₁₈₆ = anticomplement of X₂₀₇₂

• X₁₈₆ = circumcircle-inverse of X₄

• X₁₈₆ = polar-circle-inverse of X₅

• X₁₈₆ = Kosnita-circle-inverse of X₃

• X₁₈₆ = de-Longchamps-circle-inverse of X₃₇₄₄₄

• X₁₈₆ = X₃₄₀-Ceva conjugate of X₃₂₃

• X₁₈₆ = X₅₀-cross conjugate of X₃₂₃

• X₁₈₆ = crosspoint of X₅₄ and X₇₄

• X₁₈₆ = crosssum of X_i and X_j for these (i,j): (5, 30), (621, 622)

• X₁₈₆ = crossdifference of every pair of points on line X₂₁₆X₆₄₇

• X₁₈₆ = anticomplementary conjugate of anticomplement of X₃₈₅₃₄

• X₁₈₆ = pole wrt polar circle of trilinear polar of X₉₄ (line X₅X₅₂₃)

• X₁₈₆ = X₄₈-isoconjugate (polar conjugate) of X₉₄

• X₁₈₆ = perspector of ABC and the reflection of the circumorthic triangle in the Euler line

• X₁₈₆ = perspector of ABC and the reflection of the Kosnita triangle in the Euler line

• X₁₈₆ = perspector of ABC and the reflection of the orthic triangle in the orthic axis

• X₁₈₆ = reflection of X₄₀₃ in the orthic axis

• X₁₈₆ = crosspoint of X₃ and X₂₉₃₁ wrt both the excentral and tangential triangles

• X₁₈₆ = homothetic center of circumorthic and Kosnita triangles

• X₁₈₆ = perspector of circumconic through polar conjugates of PU(5)

• X₁₈₆ = Hofstadter 3 point

• X₁₈₆ = antigonal image of X₅₉₆₂

• X₁₈₆ = X₄₈₄-of-orthic-triangle if ABC is acute

• X₁₈₆ = Thomson-isogonal conjugate of X₁₅₁₃₁

• X₁₈₆ = Ehrmann-vertex-to-orthic similarity image of X₃₁₅₃

• X₁₈₆ = {X₃,X₄}-harmonic conjugate of X₃₅₂₀

• X₁₈₆ = intersection of the tangent to hyperbola {A, B, C,X₃,X₁₅}} at X₁₅ and the tangent to hyperbola {A, B, C,X₃,X₁₆}} at X₁₆

Notes

As a point on the Euler line, X₁₈₆ has Shinagawa coefficients \((4F, -E - 4F)\).

Let AA₁A₂, BB₁B₂, CC₁C₂ be the circumcircle-inscribed equilateral triangles used in the construction of the Trinh triangle. Let A' be the cevapoint of A₁ and A₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₈₆. (Randy Hutson, October 8, 2019

Let P and Q be circumcircle antipodes. X₁₈₆ is the Euler line intercept, other than X₂₀, of circle {X₂₀,P,Q}} for all P, Q. (Randy Hutson, August 30, 2020)

X₁₈₆ is the {X₃,X₂₄}-harmonic conjugate of X₄.