X(76) = 3rd BROCARD POINT

Trilinears

\(1/a3 : 1/b3 : 1/c3\)

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

Barycentrics

\(1/a2 : 1/b2 : 1/c2\)

Notes

Let A’ be the perspector of the A-McCay circle, and define B’ and C’ cyclically. The lines AA’, BB’, CC’ concur in X(76). (Randy Hutson, April 9, 2016)

X(76) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the de Longchamps line. The center of the inellipse is X(626) and its Brianchon point (perspector) is X(1502). (Randy Hutson, October 15, 2018)

Let A’B’C’ be the obverse triangle of X(1). Let A″B″C″ be the N-obverse triangle of X(1). Let A* be the barycentric product A’A″, and define B and C* cyclically. The lines AA*, BB*, CC* concur in X(76). (Randy Hutson, October 15, 2018)

X(76) is the barycentric multiplier for the Steiner circumellipse. (The barycentric product of X(76) and the circumcircle is the Steiner circumellipse.) (Randy Hutson, August 19, 2019)

For an artistic design generated by X(76), see X(244).

X(76) lies on these lines: 1,350 2,39 3,98 4,69 5,262 6,83 7,1240 8,668 10,75 13,299 14,298 17,303 18,302 20,3424 22,1799 25,1241 31,734 32,384 37,1218 85,226 95,96 100,767 107,2366 110,2367 115,626 141,698 148,2896 182,3406 187,3552 192,1221 251,1239 257,1926 275,276 297,343 321,561 330,1015 331,1231 333,1751 334,1089 335,871 338,599 485,491 486,492 524,598 620,1569 689,755 691,2868 693,764 761,789 799,1150 826,882 940,1509 1003,3053 1007,3090 1131,1271 1132,1270 1229,1446 1423,3403 1501,3115 1670,1677 1671,1676 1698,3097 2001,2909 2319,3500 2394,3267 3224,3225 3492,3506 3496,3512 3497,3509

X(76) is the {X(2),:ref:X(194) <X(194)>}-harmonic conjugate of X(39). For a list of other harmonic conjugates of X(76), click Tables at the top of this page.

X(76) = reflection of X(194) in X(39)

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

X(76) = isotomic conjugate of X(6)

X(76) = complement of X(194)

X(76) = anticomplement of X(39)

X(76) = circumcircle-inverse of X(5152)

X(76) = 2nd-Brocard circle-inverse of X(99)

X(76) = anticomplementary conjugate of X(2896)

X(76) = cyclocevian conjugate of isogonal conjugate of X(2916)

X(76) = cyclocevian conjugate of isotomic conjugate of X(1369)

X(76) = X(i)-Ceva conjugate of X(j) for these (i,j): (308,2), (310,75)

X(76) = cevapoint of X(i) and X(j) for these (i,j): (2,69), (6,22), (75,312), (311,343), (313,321), (339,525)

X(76) = X(i)-cross conjugate of X(j) for these (i,j): (2,264), (69,305), (141,2), (321,75), (343,69), (525,99)

X(76) = crosssum of X(669) and X(1084)

X(76) = crossdifference of every pair of points on line X(669)X(688)

X(76) = X(i)-beth conjugate of X(j) for these (i,j): (76,85), (799,348)

X(76) = pole wrt polar circle of trilinear polar of X(25) (line X(512)X(1692))

X(76) = X(48)-isoconjugate (polar conjugate) of X(25)

X(76) = X(6)-isoconjugate of X(31)

X(76) = trilinear product of PU(i) for these i: 10, 86

X(76) = barycentric product of PU(11)

X(76) = antigonal image of X(1916)

X(76) = cevapoint of polar conjugates of PU(4)

X(76) = trilinear product of vertices of 1st Brocard triangle

X(76) = trilinear product of vertices of 1st anti-Brocard triangle

X(76) = X(2)-Ceva conjugate of X(6374)

X(76) = X(384)-of-5th-Brocard-triangle

X(76) = X(6)-of-6th-Brocard-triangle

X(76) = perspector of ABC and 1st Brocard triangle

X(76) = trilinear pole of de Longchamps line

X(76) = bicentric sum of PU(159)

X(76) = PU(159)-harmonic conjugate of X(9494)

X(76) = perspector of conic {A,B,C,:ref:X(670) <X(670)>,:ref:X(689) <X(689)>,:ref:X(1978) <X(1978)>}} (isotomic conjugate of Lemoine axis.)

X(76) = X(1916) of 1st Brocard triangle

X(76) = crosspoint of X(6) and X(22) wrt both the anticomplementary and tangential triangles

X(76) = X(3094)-of-1st anti-Brocard-triangle

X(76) = trilinear product of vertices of mid-triangle of 1st Brocard and 1st anti-Brocard triangles

X(76) = perspector of ABC and cross-triangle of ABC and 3rd Brocard triangle

X(76) = trilinear product of vertices of the three anti-altimedial triangles

X(76) = Cundy-Parry Phi transform of X(98)

X(76) = Cundy-Parry Psi transform of X(511)

X(76) = barycentric product X(99)

X(76) = intersection, other than X(4), of P(1)- and U(1)-Fuhrmann circles (aka -Hagge circles)

X(76) = {X(7737),:ref:X(14023) <X(14023)>}-harmonic conjugate of X(20065)

X(76) = intersection of lines PU(1) of 1st and 2nd Ehrmann circumscribing triangles

X(76) = trilinear cube of X(2)

X(76) = barycentric square of X(75)

X(76) = trilinear product of vertices of Gemini triangle 19