Trilinear Polars and Antiparallels

We study the triangle bounded by the antiparallels to the sidelines of a given triangle ABC through the intercepts of the trilinear polar of a point P other than the centroid G. We show that this triangle is perspective with the reference triangle, and also study the condition of concurrency of the antiparallels. Finally, we also study the configuration of induced GP -lines and obtain an interesting conjugation of finite points other than G. 1. Perspector of a triangle bounded by antiparallels We use the barycentric coordinates with respect to triangle ABC throughout. Let P = (u : v : w) be a finite point in the plane of ABC , distinct from its centroid G. The trilinear polar of P is the line L : x u + y v + z w = 0, which intersects the sidelines BC , CA, AB respectively at Pa = (0 : v : −w), Pb = (−u : 0 : w), Pc = (u : −v : 0). The lines through Pa, Pb, Pc antiparallel to the respective sidelines of ABC are La : (b w − cv)x + (b − c)wy + (b − c)vz = 0, Lb : (c 2 − a)wx + (cu − aw)y + (c − a)uz = 0, Lc : (a 2 − b)vx + (a − b)uy + (av − bu)z = 0. They bound a triangle with vertices A′ = (−a(a(u − vw) + bu(w − u)− cu(u− v)) : (c − a)(av(w − u) + bu(v − w)) : (a − b)(cu(v − w) + aw(u− v))), B = ((b − c)(av(w − u) + bu(v −w)) : −b(b(v − wu) + cv(u− v)− av(v − w)) : (a − b)(bw(u− v) + cv(w − u))), C ′ = ((b − c)(cu(v − w) + aw(u− v)) : (c − a)(bw(u − v) + cv(w − u)) : −c(c(w − uv) + aw(v − w)− bw(w − u))). Publication Date: November 30, 2009. Communicating Editor: Paul Yiu.