X(95) = X(95) CEVAPOINT OF CENTROID AND CIRCUMCENTER¶
Trilinears
\(b2c2sec(B - C) : : Let A'B'C' be the symmedial triangle. Let La be the reflection of line B'C' in line BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in :ref:`X(95) <X(95)>\). (Randy Hutson, August 19, 2015) Let A’ be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B’ and C’ cyclically. The triangle A’B’C’ is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A’, B’, C’ lie on the rectangular hyperbola {X(2),:ref:X(20) <X(20)>,:ref:X(54) <X(54)>,:ref:X(69) <X(69)>,:ref:X(110) <X(110)>,:ref:X(2574) <X(2574)>,:ref:X(2575) <X(2575)>,:ref:X(2979) <X(2979)>}}. (See https://bernard-gibert.pagesperso-orange.fr/Exemples/k007.html.) Also, X(95) is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015) X(95) lies on these lines: 2,97 3,264 54,69 76,96 99,311 140,340 141,287 160,327 183,305 216,648 307,320 X(95) = isogonal conjugate of X(51) <X(51)>`
Notes
Let A’B’C’ be the symmedial triangle. Let La be the reflection of line B’C’ in line BC, and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(95). (Randy Hutson, August 19, 2015)
Let A’ be the intersection, other than A, of the circumcircle and the branch of the Lucas cubic that contains A, and define B’ and C’ cyclically. The triangle A’B’C’ is here introduced as the Lucas triangle (not to be confused with the Lucas central triangle). The vertices A’, B’, C’ lie on the rectangular hyperbola {X(2),:ref:X(20) <X(20)>,:ref:X(54) <X(54)>,:ref:X(69) <X(69)>,:ref:X(110) <X(110)>,:ref:X(2574) <X(2574)>,:ref:X(2575) <X(2575)>,:ref:X(2979) <X(2979)>}}. (See https://bernard-gibert.pagesperso-orange.fr/Exemples/k007.html.) Also, X(95) is the trilinear product of the vertices of the Lucas triangle. (Randy Hutson, August 19, 2015)
X(95) lies on these lines: 2,97 3,264 54,69 76,96 99,311 140,340 141,287 160,327 183,305 216,648 307,320
X(95) = isogonal conjugate of X(51)
X(95) = isotomic conjugate of X(5)
X(95) = anticomplement of X(233)
X(95) = X(276)-Ceva conjugate of X(275)
X(95) = cevapoint of X(i) and X(j) for these (i,j): (2,3), (6,160), (54,97)
X(95) = X(i)-cross conjugate of X(j) for these (i,j): (2,276), (3,97), (54,275), (140,2), (340,1494)
X(95) = intersection of tangents at X(2) and X(3) to bianticevian conic of X(2) and X(3)
X(95) = crosspoint of X(2) and X(3) wrt both the anticomplementary triangle and anticevian triangle of X(3)
X(95) = trilinear pole of line X(323)X(401) (polar of X(53) wrt polar circle, and polar of X(69) wrt de Longchamps circle)
X(95) = pole wrt polar circle of trilinear polar of X(53)
X(95) = X(48)-isoconjugate (polar conjugate) of X(53)
X(95) = X(92)-isoconjugate of X(217)