Problem

Source: Brazilian Mathematical Olympiad 2023, Level 3, Problem 2

Tags: geometry, right triangle, Equilateral Triangle



Let $ABC$ be a right triangle in $B$, with height $BT$, $T$ on the hypotenuse $AC$. Construct the equilateral triangles $BTX$ and $BTY$ so that $X$ is in the same half-plane as $A$ with respect to $BT$ and $Y$ is in the same half-plane as $C$ with respect to $BT$. Point $P$ is the intersection of $AY$ and $CX$. Show that $$PA \cdot BC = PB \cdot CA = PC \cdot AB.$$