A New Characterization of the Symmedian Point
Year: 2026
Type: article
Abstract: 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.
Source: Geometry
Authors: Yu Chen, Robert J. Fisher
Institution: Idaho State University
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,207
OpenAlex ID: W7119504101