X₃₈₄₅

X₃₈₁com(EULER TRIANGLE)

X₃₈₄₅ = orthocentroidal-circle-inverse of X₃₈₃₀

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

X₃₈₄₅ (X₃₈₁com(EULER TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 1327}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₁₃₂₇ (ARCTAN(3) KIEPERT POINT)

Barycentrics: \(4a^{4} - 5b^{4} - 5c^{4} + a^{2}b^{2} + a^{2}c^{2} + 10b^{2}c^{2} : :\)

Search Value (6, 9, 13): \(-3.600122360073\)

▸ Linear Combinations1 combination

• X₃₈₄₅ = X₂ + \(3\) X₄

▸ Incident Central Lines10 lines

{2, 3}, {6, 1327}, {343, 1531}, {399, 1994}, {495, 3058}, {496, 3585}, {541, 1539}, {542, 1353}, {946, 1483}, {952, 1699}

▸ Functional Relations9 relations

• X₃₈₄₅ = circumcircle-inverse of X₃₇₉₅₈

• X₃₈₄₅ = orthocentroidal-circle-inverse of X₃₈₃₀

• X₃₈₄₅ = Ehrmann-side-to-orthic similarity image of X₂

• X₃₈₄₅ = Johnson-to-Ehrmann-mid similarity image of X₃₈₁

• X₃₈₄₅ = trisector nearest X₄ of segment X₄X₅

• X₃₈₄₅ = X₂-of-Ehrmann-mid-triangle

• X₃₈₄₅ = X₁₀₂₄₆-of-orthic-triangle if ABC is acute

• X₃₈₄₅ = {X₂,X₄}-harmonic conjugate of X₃₈₃₀

• X₃₈₄₅ = {X₁₃₅₉₅,X₁₃₅₉₆}-harmonic conjugate of X₃

Notes

As a point on the Euler line, X₃₈₄₅ has Shinagawa coefficients \((1, 9)\).

X₃₈₄₅ is the point P which divides segment X₄X₅ in the ratio PX₄/PX₅ = -1/2. (Randy Hutson, June 27, 2018)