Problem

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 8.4

Tags: geometry, rhombus, tangency



Point $T$ is chosen in the plane of a rhombus $ABCD$ so that $\angle ATC + \angle BTD = 180^\circ$, and circumcircles of triangles $ATC$ and $BTD$ are tangent to each other. Show that $T$ is equidistant from diagonals of $ABCD$. Proposed by Fedir Yudin