Three Properties of the Symmedian Point / Darij Grinberg 1. on Isogonal Conjugates

Fig. 1 The de…nition of isogonal conjugates is based on the following theorem (Fig. 1): Theorem 1. Let ABC be a triangle and P a point in its plane. Then, the re‡ections of the lines AP; BP; CP in the angle bisectors of the angles CAB; ABC; BCA concur at one point. This point is called the isogonal conjugate of the point P with respect to triangle ABC: We denote this point by Q: Note that we work in the projective plane; this means that in Theorem 1, both the point P and the point of concurrence of the re‡ections of the lines AP; BP; CP in the angle bisectors of the angles CAB; ABC; BCA can be in…nite points.