X₃₂

3rd POWER POINT

X₃₂ = Brocard-circle-inverse of X₃₉

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

X₃₂ (3rd POWER POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {371, 372}

• Related Centers: X₃₇₁ (KENMOTU POINT), X₃₇₂ ({X)

Trilinears: \(a^{3} : b^{3} : c^{3}\)

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

Search Value (6, 9, 13): \(0.280712446186\)

▸ Alternate Coordinates6 forms

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

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

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

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

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

• Trilinears: \(a - 2R \cos A \tan \omega : :\)

▸ Linear Combinations1 combination

• X₃₂ = \(-(r^{2} + 4rR - s^{2})(r^{2} + 2rR + s^{2})\) X₁ + \(6rR(r^{2} + 4rR - s^{2})\) X₂ + \(2r^{2}(r^{2} + 4rR - 3s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines84 lines

{1, 172}, {2, 83}, {3, 6}, {4, 98}, {5, 230}, {9, 987}, {20, 2549}, {21, 981}, {22, 1194}, {24, 232}, {25, 1184}, {31, 41}, {35, 2276}, {48, 1472}, {51, 2351}, {55, 1500}, {56, 1015}, {71, 2273}, {75, 746}, {76, 384}, {81, 980}, {99, 194}, {100, 713}, {101, 595}, {110, 729}, {111, 1383}, {163, 849}, {165, 1571}, {184, 211}, {218, 906}, {220, 3052}, {262, 3406}, {263, 1976}, {512, 878}, {538, 1003}, {560, 1918}, {561, 724}, {590, 640}, {604, 1106}, {615, 639}, {632, 3055}, {637, 3069}, {638, 3068}, {682, 1974}, {695, 3492}, {710, 1502}, {731, 825}, {733, 827}, {902, 1334}, {904, 1933}, {910, 1104}, {941, 1169}, {958, 1572}, {983, 3495}, {988, 1449}, {993, 1107}, {1009, 1724}, {1055, 1201}, {1084, 1576}, {1092, 3289}, {1191, 3207}, {1204, 3269}, {1376, 1574}, {1395, 1402}, {1423, 3500}, {1468, 2280}, {1613, 1915}, {1843, 2353}, {1911, 1932}, {1919, 3249}, {1922, 1923}, {1950, 2285}, {1951, 2082}, {1992, 2482}, {1995, 3291}, {2004, 2005}, {2319, 3494}, {2508, 2881}, {2698, 2715}, {3087, 3088}, {3124, 3457}, {3170, 3171}, {3497, 3512}, {3499, 3511}

▸ Functional Relations59 relations

• X₃₂ = midpoint of X₃₇₁ and X₃₇₂

• X₃₂ = reflection of X₃₁₅ in X₆₂₆

• X₃₂ = isogonal conjugate of X₇₆

• X₃₂ = isotomic conjugate of X₁₅₀₂

• X₃₂ = complement of X₃₁₅

• X₃₂ = anticomplement of X₆₂₆

• X₃₂ = circumcircle-inverse of X₁₆₉₁

• X₃₂ = Brocard-circle-inverse of X₃₉

• X₃₂ = 1st-Lemoine-circle-inverse of X₁₆₉₂

• X₃₂ = antigonal conjugate of X₃₇₈₄₁

• X₃₂ = anticomplementary conjugate of anticomplement of X₃₈₈₂₆

• X₃₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 206), (6, 184), (112, 512), (251, 6)

• X₃₂ = crosspoint of X_i and X_j for these (i,j): (2, 66), (6, 25)

• X₃₂ = crosssum of X_i and X_j for these (i,j): (2, 69), (6, 22), (75, 312), (115, 826), (311, 343), (313, 321), (338, 850), (339, 525), (349, 1231), (693, 1086), (1229, 1233), (1230, 1269)

• X₃₂ = crossdifference of every pair of points on line X₃₂₅X₅₂₃

• X₃₂ = X₁₈₄-Hirst inverse of X₂₃₇

• X₃₂ = X_i-beth conjugate of X_j for these (i,j): (41, 41), (163, 56), (919, 32)

• X₃₂ = external center of similitude of circumcircle and Moses circle

• X₃₂ = radical trace of circumcircle and circle {X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}

• X₃₂ = trilinear product of vertices of circumsymmedial triangle

• X₃₂ = trilinear product of vertices of 3rd Brocard triangle

• X₃₂ = insimilicenter of circles O(15,16) and O(61,62); the exsimilicenter is X₃₉

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

• X₃₂ = intersection of tangents at PU(1) to Brocard circle

• X₃₂ = intersection of lines P(1)U(2) and U(1)P(2)

• X₃₂ = vertex conjugate of PU(1)

• X₃₂ = trilinear product of PU(9)

• X₃₂ = barycentric product of PU(36)

• X₃₂ = bicentric sum of PU(39)

• X₃₂ = midpoint of PU(39)

• X₃₂ = center of circle {X₃₇₁,X₃₇₂,PU(1),PU(39)}} (the circle orthogonal to the Brocard circle through the 1st and 2nd Brocard points)

• X₃₂ = crosssum of polar conjugates of PU(4)

• X₃₂ = perspector ABC and tangential triangle of 1st Brocard triangle

• X₃₂ = trilinear cube of X₆

• X₃₂ = trilinear square root of X₁₉₁₇

• X₃₂ = inverse-in-2nd-Brocard-circle of X₃₀₉₄

• X₃₂ = perspector of circumconic centered at X₂₀₆

• X₃₂ = center of circumconic that is locus of trilinear poles of lines passing through X₂₀₆

• X₃₂ = trilinear pole of line X₆₆₉X₆₈₈ (the isogonal conjugate of the isotomic conjugate of the Lemoine axis)

• X₃₂ = perspector of ABC and 3rd Brocard triangle

• X₃₂ = {X₆₁,X₆₂}-harmonic conjugate of X₅₇₆

• X₃₂ = {X₁₃₄₀,X₁₃₄₁}-harmonic conjugate of X₅₁₁₆

• X₃₂ = {X₁₆₈₇,X₁₆₈₈}-harmonic conjugate of X₃

• X₃₂ = reflection of X₅₀₂₈ in X₆

• X₃₂ = X₃₂-of-circumsymmedial-triangle

• X₃₂ = X₇₅-isoconjugate of X₂

• X₃₂ = X₉₂-isoconjugate of X₆₉

• X₃₂ = X₁₅₇₇-isoconjugate of X₉₉

• X₃₂ = X₄₀₄₈ of 1st anti-Brocard triangle

• X₃₂ = homothetic center of circumnormal triangle and unary cofactor triangle of Stammler triangle

• X₃₂ = perspector of ABC and cross-triangle of ABC and 1st Brocard triangle

• X₃₂ = homothetic center of medial triangle and cross-triangle of ABC and 5th Brocard triangle

• X₃₂ = homothetic center of medial triangle and cross-triangle of ABC and 5th anti-Brocard triangle

• X₃₂ = Cundy-Parry Phi transform of X₅₁₁

• X₃₂ = Cundy-Parry Psi transform of X₉₈

• X₃₂ = X₁₆₉-of-orthic-triangle if ABC is acute

• X₃₂ = Steiner-circumellipse-inverse of X₁₆₉₈₅

• X₃₂ = barycentric square of X₆

• X₃₂ = barycentric product of (nonreal) circumcircle intercepts of the Lemoine axis

Notes

The 5th Brocard triangle is here introduced as the vertex triangle of the circumcevian triangles of the 1st and 2nd Brocard points. (Randy Hutson, December 26, 2015)

The 5th Brocard triangle is homothetic to ABC at X₃₂, homothetic to the medial triangle at X₃₀₉₆, homothetic to the anticomplementary triangle at X₂₈₉₆, perspective to the 1st Brocard triangle at X₂₈₉₆, and perspective to the 3rd Brocard triangle at X₃₂.(Randy Hutson, December 26, 2015)

Let A'B'C' be the 1st Brocard triangle. Let A″, B″, C″ be inverse-in-circumcircle of A', B' and C' resp. AA″, BB″, CC″ concur in X₃₂. (Randy Hutson, July 20, 2016)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the cevapoint, wrt A'B'C', of B and C, and define B″, C″ cyclically. A'A″, B'B″, C'C″ concur in X₃₂. (Randy Hutson, July 20, 2016)

X₃₂ is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X₈₂₆₅. (Randy Hutson, October 15, 2018)

X₃₂ is the {X₃,X₆}-harmonic conjugate of X₃₉.