AoPS Wiki talk:Problem of the Day/June 19, 2011
Let the angles of the triangles at the interior point be and , such that . Assume the contrary, that there are at least acute triangles. Assume WLOG that and are acute or right angles, so that . Therefore, . Now, since , we have , so . However, this is a contradiction, since must be the vertex of a triangle, and therefore cannot be more than . Therefore, there cannot be more than acute or right angles at the interior point, and therefore there must be at least obtuse angles, creating at least obtuse triangles.