X₁₆₇₀

INVERSE-IN-BROCARD-CIRCLE OF X₁₃₄₂

X₁₆₇₀ is the external center of similitude of the Gallatly circle and the 2nd Lemoine circle. (Peter J. C. Moses, 9/03; cf. X₁₃₄₂)

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

X₁₆₇₀ (INVERSE-IN-BROCARD-CIRCLE OF X₁₃₄₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1671, 3}

• Related Centers: X₁₆₇₁ (INVERSE-IN-BROCARD-CIRCLE OF X₁₃₄₃), X₃ (CIRCUMCENTER)

Trilinears: \(\sin A - \cos A \cot(\omega /2) : \sin B - \cos B \cot(\omega /2) : \sin C - \cos C \cot(\omega /2)\)

Search Value (6, 9, 13): \(12.278606804608\)

▸ Alternate Coordinates5 forms

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

• Trilinears: \(\cos A + \cos(A + \omega ) : \cos B + \cos(B + \omega ) : \cos C + \cos(C + \omega )\)

• Trilinears: \(\cos(A + \omega /2) : \cos(B + \omega /2) : \cos(C + \omega /2)\)

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

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

▸ Incident Central Lines11 lines

{3, 6}, {76, 1677}, {262, 1676}, {485, 2009}, {486, 2010}, {1124, 2007}, {1335, 2008}, {1377, 2013}, {1378, 2014}, {1702, 2017}, {1703, 2018}

▸ Functional Relations10 relations

• X₁₆₇₀ = reflection of X₁₆₇₁ in X₃

• X₁₆₇₀ = isogonal conjugate of X₁₆₇₇

• X₁₆₇₀ = circumcircle-inverse of X₃₈₇₂₀

• X₁₆₇₀ = Brocard-circle-inverse of X₁₃₄₂

• X₁₆₇₀ = X₇₆-Ceva conjugate of X₁₆₇₁

• X₁₆₇₀ = Thomson-isogonal conjugate of X₃₃₇₀₇

• X₁₆₇₀ = {X₆,X₃₉}-harmonic conjugate of X₁₆₇₁

• X₁₆₇₀ = {X₃₂,X₃₀₉₄}-harmonic conjugate of X₁₆₇₁

• X₁₆₇₀ = {X₁₈₂,X₃₀₉₅}-harmonic conjugate of X₁₆₇₁

• X₁₆₇₀ = {X₁₃₇₉,X₁₃₈₀}-harmonic conjugate of X₃₈₇₂₀

Notes

X₁₆₇₀ and X₁₆₇₁ are the Brocard axis intercepts of the 2nd Brocard circle. (Randy Hutson, August 29, 2018)