A New Characterization of the Symmedian Point

Let ▵ABC be a triangle in the plane E, K be its symmedian point, and C be its circumcircle. Assume that P is a point on C such that it is not A, B, or C, it does not lie on the medians of ▵ABC, and the lines ℓAP, ℓBP, and ℓCP intersect ℓBC, ℓCA, and ℓAB at points Pa, Pb, and Pc, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates Qa, Qb, and Qc of Pa, Pb, and Pc in BC¯, CA¯, and AB¯, respectively, are collinear. We prove that K lies on the line through Qa, Qb, and Qc, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle C onto the set LK of all the lines in E that pass through K.