Crux 5145: An Isogonal Concurrency Problem

Problem

Recall that a pair of lines passing through the vertex of a given angle are called isogonal if they make equal angles with the bisector of the angle. For a given triangle $ABC$ , define $X_1$ and $X_2$ to be a pair of points on $BC$ for which the lines $AX_1$ and $AX_2$ are isogonal, and similarly for $Y_1$ and $Y_2$ on $CA$ and for $Z_1$ and $Z_2$ on $AB$ . Finally define

\begin{aligned} P&=Z_2X_1\cap X_2Y_1,\\ Q&=X_2Y_1\cap Y_2Z_1,\\ R&=Y_2Z_1\cap Z_2X_1. \end{aligned}

Prove that the lines $AP,BQ,CR$ are either concurrent or parallel.

Publication status

This problem was proposed by Chikara Tsugawa and published as Problem 5145 in Crux Mathematicorum, Vol. 52(5), May 2026, p. 242. The solution is intentionally withheld until the journal publishes the corresponding solutions. This page will then be updated with the proposer’s own solution and the relevant publication record.