X(32) = 3rd POWER POINT

Trilinears

\(a3 : b3 : c3\)

\(sin(A - ω) : sin(B - ω) : sin(C - ω)\)

\(sin A + sin(A - 2ω) : sin B + sin(B - 2ω) : sin C + sin(C - 2ω)\)

\(cos A - cos(A - 2ω) : cos B - cos(B - 2ω) : cos C - cos(C -2ω)\)

\(cos A - sin A cot ω : :\)

\(sin A - cos A tan ω : :\)

\(a - 2R cos A tan ω : :\)

Barycentrics

\(a4 : b4 : c4\)

Notes

If you have The Geometer’s Sketchpad, you can view X(32).

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 X(32), homothetic to the medial triangle at X(3096), homothetic to the anticomplementary triangle at X(2896), perspective to the 1st Brocard triangle at X(2896), and perspective to the 3rd Brocard triangle at X(32).(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 X(32). (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 X(32). (Randy Hutson, July 20, 2016)

X(32) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X(8265). (Randy Hutson, October 15, 2018)

X(32) lies on these 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

X(32) is the {X(3),:ref:X(6) <X(6)>}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(32), click Tables at the top of this page.

X(32) = midpoint of X(371) and X(372)

X(32) = reflection of X(315) in X(626)

X(32) = isogonal conjugate of X(76)

X(32) = isotomic conjugate of X(1502)

X(32) = complement of X(315)

X(32) = anticomplement of X(626)

X(32) = circumcircle-inverse of X(1691)

X(32) = Brocard-circle-inverse of X(39)

X(32) = 1st-Lemoine-circle-inverse of X(1692)

X(32) = antigonal conjugate of X(37841)

X(32) = anticomplementary conjugate of anticomplement of X(38826)

X(32) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,206), (6,184), (112,512), (251,6)

X(32) = crosspoint of X(i) and X(j) for these (i,j): (2,66), (6,25)

X(32) = 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)

X(32) = crossdifference of every pair of points on line X(325)X(523)

X(32) = X(184)-Hirst inverse of X(237)

X(32) = X(i)-beth conjugate of X(j) for these (i,j): (41,41), (163,56), (919,32)

X(32) = external center of similitude of circumcircle and Moses circle

X(32) = radical trace of circumcircle and circle {X(1687),:ref:X(1688) <X(1688)>,PU(1),PU(2)}

X(32) = trilinear product of vertices of circumsymmedial triangle

X(32) = trilinear product of vertices of 3rd Brocard triangle

X(32) = insimilicenter of circles O(15,16) and O(61,62); the exsimilicenter is X(39)

X(32) = insimilicenter of circles {X(15),:ref:X(62) <X(62)>,PU(1)} and {X(16),:ref:X(61) <X(61)>,PU(1)}; the exsimilicenter is X(182)

X(32) = intersection of tangents at PU(1) to Brocard circle

X(32) = intersection of lines P(1)U(2) and U(1)P(2)

X(32) = vertex conjugate of PU(1)

X(32) = trilinear product of PU(9)

X(32) = barycentric product of PU(36)

X(32) = bicentric sum of PU(39)

X(32) = midpoint of PU(39)

X(32) = center of circle {X(371),:ref:X(372) <X(372)>,PU(1),PU(39)}} (the circle orthogonal to the Brocard circle through the 1st and 2nd Brocard points)

X(32) = crosssum of polar conjugates of PU(4)

X(32) = perspector ABC and tangential triangle of 1st Brocard triangle

X(32) = trilinear cube of X(6)

X(32) = trilinear square root of X(1917)

X(32) = inverse-in-2nd-Brocard-circle of X(3094)

X(32) = perspector of circumconic centered at X(206)

X(32) = center of circumconic that is locus of trilinear poles of lines passing through X(206)

X(32) = trilinear pole of line X(669)X(688) (the isogonal conjugate of the isotomic conjugate of the Lemoine axis)

X(32) = perspector of ABC and 3rd Brocard triangle

X(32) = {X(61),:ref:X(62) <X(62)>}-harmonic conjugate of X(576)

X(32) = {X(1340),:ref:X(1341) <X(1341)>}-harmonic conjugate of X(5116)

X(32) = {X(1687),:ref:X(1688) <X(1688)>}-harmonic conjugate of X(3)

X(32) = reflection of X(5028) in X(6)

X(32) = X(32)-of-circumsymmedial-triangle

X(32) = X(75)-isoconjugate of X(2)

X(32) = X(92)-isoconjugate of X(69)

X(32) = X(1577)-isoconjugate of X(99)

X(32) = X(4048) of 1st anti-Brocard triangle

X(32) = homothetic center of circumnormal triangle and unary cofactor triangle of Stammler triangle

X(32) = perspector of ABC and cross-triangle of ABC and 1st Brocard triangle

X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th Brocard triangle

X(32) = homothetic center of medial triangle and cross-triangle of ABC and 5th anti-Brocard triangle

X(32) = Cundy-Parry Phi transform of X(511)

X(32) = Cundy-Parry Psi transform of X(98)

X(32) = X(169)-of-orthic-triangle if ABC is acute

X(32) = Steiner-circumellipse-inverse of X(16985)

X(32) = barycentric square of X(6)

X(32) = barycentric product of (nonreal) circumcircle intercepts of the Lemoine axis