X₁₈₂

MIDPOINT OF BROCARD DIAMETER

X₁₈₂ is the midpoint of the Brocard diameter (the segment X₃-to-X₆); also the center of the 1st Lemoine circle, and the center of the Brocard circle.

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

X₁₈₂ (MIDPOINT OF BROCARD DIAMETER) 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: \(\cos(A- \omega ) : \cos(B - \omega ) : \cos(C -\omega )\)

Barycentrics: \(\sin A \cos(A - \omega ) : \sin B \cos(B - \omega ) : \sin C \cos(C -\omega )\)

Search Value (6, 9, 13): \(3.887635694631\)

▸ Alternate Coordinates7 forms

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

• Trilinears: \(\sin A - \sin(A - 2\omega ) : \sin B - \sin(B - 2\omega ) : \sin C - \sin(C - 2\omega )\)

• Trilinears: cos A + cos(A - 2ω) : cos B + cos(B - 2ω) : cos C + cos(C - 2ω) (cf., X₃₉)

• Trilinears: a + 2R cot ω cos A : b + 2R cot ω cos B: c + 2R cot ω cos C (cf., X₁₃₅₀, X₁₃₅₁)

• Trilinears: sin A + cos A cot ω : sin B + cos B cot ω : sin C + cos C cot ω (cf., X₅₇₅, X₅₇₆,,X₁₃₅₀, X₁₃₅₁)

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

• Barycentrics: \(a^{2}(a^{4} - a^{2}b^{2} - a^{2}c^{2} - 2b^{2}c^{2}) : :\)

▸ Linear Combinations1 combination

• X₁₈₂ = X₃ + X₆

▸ Incident Central Lines28 lines

{1, 983}, {2, 98}, {3, 6}, {4, 83}, {5, 206}, {10, 1678}, {22, 51}, {24, 1843}, {25, 3066}, {30, 597}, {36, 1469}, {40, 1700}, {54, 69}, {55, 613}, {56, 611}, {111, 353}, {140, 141}, {171, 1397}, {373, 1495}, {474, 1437}, {517, 1386}, {518, 1385}, {524, 549}, {691, 2698}, {692, 1001}, {727, 1293}, {729, 1296}, {952, 996}

▸ Functional Relations41 relations

• X₁₈₂ = radical center of Lucas(2 tan ω) circles, where 2 tan ω is the value of t for which the Brocard circle is the radical circle of the Lucas(t) circles. (Randy Hutson, January 29, 2015)

• X₁₈₂ = midpoint of X₃ and X₆

• X₁₈₂ = reflection of X_i in X_j for these (i,j): (6, 575), (141, 140), (576, 6)

• X₁₈₂ = isogonal conjugate of X₂₆₂

• X₁₈₂ = isotomic conjugate of X₃₂₇

• X₁₈₂ = complement of X₁₃₅₂

• X₁₈₂ = X₃-of-1st-Brocard triangle

• X₁₈₂ = X₃- of 2nd Brocard triangle

• X₁₈₂ = X₁₈₂-of-circumsymmedial triangle

• X₁₈₂ = {X₃,X₆}-harmonic conjugate of X₅₁₁

• X₁₈₂ = {X₆,X₁₃₅₀}-harmonic conjugate of X₁₃₅₁

• X₁₈₂ = {X₁₃₄₀,X₁₃₄₁}-harmonic conjugate of X₃

• X₁₈₂ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₆

• X₁₈₂ = vertex conjugate of PU(191)

• X₁₈₂ = X₅-of-obverse-triangle-of-X₆₉

• X₁₈₂ = inverse-in-circumcircle of X₂₀₈₀

• X₁₈₂ = inverse-in-2nd-Brocard-circle of X₃₀₉₅

• X₁₈₂ = inverse-in-circle-{X₃₁₀₂,X₃₁₀₃,PU(1)}} of X₃₂₄₅₂

• X₁₈₂ = exsimilicenter of circle centered at X₃₇₁ through X₁₁₅₁ and circle centered at X₁₁₅₂ through X₃₇₂

• X₁₈₂ = exsimilicenter of circle centered at X₃₇₂ through X₁₁₅₂ and circle centered at X₁₁₅₁ through X₃₇₁

• X₁₈₂ = radical trace of circles with diameters X₃₇₁X₃₇₂ and X₁₁₅₁X₁₁₅₂

• X₁₈₂ = harmonic center of 1st and 2nd Kenmotu circles

• X₁₈₂ = {X₁₅,X₁₆}-harmonic conjugate of X₅₇₄

• X₁₈₂ = harmonic center of Lucas radical circle and Lucas(-1) radical circle

• X₁₈₂ = harmonic center of Lucas inner circle and Lucas(-1) inner circle

• X₁₈₂ = harmonic center of 2nd Lemoine circle and circle {X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}}

• X₁₈₂ = radical trace of circles O(15,16) and O(61,62)

• X₁₈₂ = exsimilicenter of circles {X₁₅,X₆₂,PU(1)} and {X₁₆,X₆₁,PU(1)}; the insimilicenter is X₃₂

• X₁₈₂ = X₂-of-1st-Ehrmann-triangle

• X₁₈₂ = {X₉₇₃₈,X₉₇₃₉}-harmonic conjugate of X₉₇₃₇

• X₁₈₂ = Artzt-to-McCay similarity image of X₃₈₁

• X₁₈₂ = X₃-of-6th-anti-Brocard-triangle

• X₁₈₂ = X₅₄₇₆-of-4th-anti-Brocard-triangle

• X₁₈₂ = homothetic center of 5th anti-Brocard triangle and cevian triangle of X₃

• X₁₈₂ = homothetic center of 6th anti-Brocard triangle and 1st Brocard triangle

• X₁₈₂ = endo-homothetic center of 6th Brocard triangle and 1st anti-Brocard triangle

• X₁₈₂ = perspector of 1st Neuberg triangle and cross-triangle of 1st and 2nd Neuberg triangles

• X₁₈₂ = Cundy-Parry Phi transform of X₃₉

• X₁₈₂ = Cundy-Parry Psi transform of X₈₃

• X₁₈₂ = endo-similarity image of reflection triangles of PU(1); the similitude center of these triangles is X₆

• X₁₈₂ = 1st-Brocard-isogonal conjugate of X₁₃₅₂

Notes

X₁₈₂ is the {X₃₇₁,X₃₇₂}-harmonic conjugate of X₃₉.