X76¶
3rd BROCARD POINT
X76 (3rd BROCARD POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶
Constructed Elements
Central Lines: {194, 39}
Related Centers: X194 (-CEVA CONJUGATE OF X2), X39 (BROCARD MIDPOINT)
Alternate Coordinates1 form
Trilinears: \(\csc(A - \omega ) : \csc(B - \omega ) : \csc(C - \omega )\)
Incident Central Lines69 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}
Functional Relations48 relations
X76 = isogonal conjugate of X32
X76 = isotomic conjugate of X6
X76 = complement of X194
X76 = circumcircle-inverse of X5152
X76 = cyclocevian conjugate of isogonal conjugate of X2916
X76 = Xi-Ceva conjugate of Xj for these (i,j): (308, 2), (310, 75)
X76 = cevapoint of Xi and Xj for these (i,j): (2, 69), (6, 22), (75, 312), (311, 343), (313, 321), (339, 525)
X76 = Xi-cross conjugate of Xj for these (i,j): (2, 264), (69, 305), (141, 2), (321, 75), (343, 69), (525, 99)
X76 = crossdifference of every pair of points on line X669X688
X76 = Xi-beth conjugate of Xj for these (i,j): (76, 85), (799, 348)
X76 = pole wrt polar circle of trilinear polar of X25 (line X512X1692)
X76 = X48-isoconjugate (polar conjugate) of X25
X76 = trilinear product of PU(i) for these i: 10, 86
X76 = barycentric product of PU(11)
X76 = antigonal image of X1916
X76 = cevapoint of polar conjugates of PU(4)
X76 = trilinear product of vertices of 1st Brocard triangle
X76 = trilinear product of vertices of 1st anti-Brocard triangle
X76 = perspector of ABC and 1st Brocard triangle
X76 = trilinear pole of de Longchamps line
X76 = bicentric sum of PU(159)
X76 = PU(159)-harmonic conjugate of X9494
X76 = perspector of conic {A, B, C,X670,X689,X1978}} (isotomic conjugate of Lemoine axis.)
X76 = crosspoint of X6 and X22 wrt both the anticomplementary and tangential triangles
X76 = trilinear product of vertices of mid-triangle of 1st Brocard and 1st anti-Brocard triangles
X76 = perspector of ABC and cross-triangle of ABC and 3rd Brocard triangle
X76 = trilinear product of vertices of the three anti-altimedial triangles
X76 = intersection, other than X4, of P(1)- and U(1)-Fuhrmann circles (aka -Hagge circles)
X76 = {X7737,X14023}-harmonic conjugate of X20065
X76 = intersection of lines PU(1) of 1st and 2nd Ehrmann circumscribing triangles
X76 = trilinear product of vertices of Gemini triangle 19
Notes¶
Let A' be the perspector of the A-McCay circle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X76. (Randy Hutson, April 9, 2016)
X76 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 X626 and its Brianchon point (perspector) is X1502. (Randy Hutson, October 15, 2018)
Let A'B'C' be the obverse triangle of X1. Let A″B″C″ be the N-obverse triangle of X1. Let A* be the barycentric product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X76. (Randy Hutson, October 15, 2018)
X76 is the barycentric multiplier for the Steiner circumellipse. (The barycentric product of X76 and the circumcircle is the Steiner circumellipse.) (Randy Hutson, August 19, 2019)