KöMaL problem A, 2025

Given a circle with three points $A$, $B$, and $C$ on it, which do not form an isosceles triangle. For every point $P\notin\{A,B,C\}$ on the circle, let $A_P$, $B_P$ and $C_P$ denote the intersections of the tangent at $P$ with the tangents at $A$, $B$, and $C$, respectively. Prove that there exist exactly three points $P\notin\{A,B,C\}$ on the circle for which the points $A_P$, $B_P$ and $C_P$ are well-defined and the perpendiculars from $A_P$ to $BC$, from $B_P$ to $CA$, and from $C_P$ to $AB$ are concurrent. Furthermore, show that these three points $P$ form an equilateral triangle.