JMO 1996 Problem 1

A plane is partitioned into triangles. Let $\mathcal{T}_0$ denote the set of vertices of triangles in the partition. Let $ABC$ be a triangle with $A,B,C\in\mathcal{T}_0$ and $\theta$ be its smallest angle. Assume that no point of $\mathcal{T}_0$ lies inside the circumcircle of $\triangle ABC$. Prove that there exists a triangle $\sigma$ in the partition such that its intersection with $\triangle ABC$ is nonempty and whose every angle is greater than $\theta$.