X₁₃₄₂

INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

X₁₃₄₂ is the insimilicenter(circumcircle, 1stlemoine circle).

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

X₁₃₄₂ (INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT)

Trilinears: \(\sin A + \cos A \cot(\omega /2) : :\)

Search Value (6, 9, 13): \(4.886262128935\)

▸ Alternate Coordinates5 forms

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

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

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

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

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

▸ Incident Central Lines1 line

{3, 6}

▸ Functional Relations8 relations

• X₁₃₄₂ =(sec ω)*X₃ + 2*X₁₈₂ = (1 + sec ω)*X₃ + X₆

• X₁₃₄₂ = internal center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003; cf. X₁₃₄₃, X₁₆₇₀, X₁₆₇₁)

• X₁₃₄₂ = reflection of X₁₃₄₃ in X₃₃₉₈

• X₁₃₄₂ = isogonal conjugate of X₅₄₀₃

• X₁₃₄₂ = Brocard-circle-inverse of X₁₆₇₀

• X₁₃₄₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (371, 372, 1671), (1687, 1688, 1343)

• X₁₃₄₂ = insimilicenter of 1st and 2nd Brocard circles; the insimilicenter is X₁₃₄₃

• X₁₃₄₂ = inner-Montesdeoca-Lemoine-circle-inverse of X₃₈₇₂₀