X₁₃₄₃

EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

X₁₃₄₃ is the exsimilicenter(circumcircle, 1stlemoine circle).

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

X₁₃₄₃ (EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1342, 3398}

• Related Centers: X₁₃₄₂ (INSIMILICENTER), X₃₃₉₈ (COS(A - 2ω) POINT)

Trilinears: \(\sin A - \cos A \tan(\omega /2) : :\)

Search Value (6, 9, 13): \(0.666650980227\)

▸ Alternate Coordinates6 forms

• Trilinears: \(\sin A + \sin(A - \omega ) : :\)

• Trilinears: \(\cos A - \cos(A - \omega ) : :\)

• Trilinears: \(\sin(A - \omega /2) : :\)

• Trilinears: \(\sin A - (\sec \omega + \tan \omega ) \cos A: :\)

• Trilinears: \(\sin A + (\cot \omega - \csc \omega ) \cos A: :\)

• Trilinears: \(\cos A - (\csc \omega + \cot \omega ) \sin A : :\)

▸ Linear Combinations1 combination

• X₁₃₄₃ = \((\sec \omega )\) X₃ - \(2\) X₁₈₂ = \((1 - \sec \omega )\) X₃ + X₆

▸ Incident Central Lines1 line

{3, 6}

▸ Functional Relations7 relations

• X₁₃₄₃ = external center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₃ = reflection of X₁₃₄₂ in X₃₃₉₈

• X₁₃₄₃ = isogonal conjugate of X₅₄₀₄

• X₁₃₄₃ = inverse-in-Brocard-circle of X₁₆₇₁

• X₁₃₄₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (371, 372, 1670), (1687, 1688, 1342)

• X₁₃₄₃ = exsimilicenter of 1st and 2nd Brocard circles (the insimilicenter is X₁₃₄₂)

• X₁₃₄₃ = outer-Montesdeoca-Lemoine-circle-inverse of X₃₈₇₂₁

Notes

Let Lbc be the line obtained by rotating line CA through C by an angle of ω/2 toward B. Let Lcb be the line obtained by rotating line AB through B by an angle of ω/2 toward C. Let A' =Lbc∩Lcb. Define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₃₄₃. (Randy Hutson, October 13, 2015)