## Titeica 3 circle problem (5 Lei problem)

It is said that the Romanian Mathematicican Gheorghe Titeica discovered it accidentally while drawing circles with a 5-lei Romanian coin in 1908, and proposed it the same year at a competition organized by Romanian Mathematical Gazette.

Prove that three distinct equal radius circles which have a common point meet at other three points which have the property that the circumcircle of the triangle determined by these three has the same radius as the three initial circles. (You can check this right away using a coin 😉 )

**Proof:** There are several proofs of this theorem. First, consider drawing the centers of the three circles. A number of rhombuses appear, and using triangle congruences it should be a simple task.

A more intriguing proof, due to Polya and Szego uses the fact that the figure determined by the three points of intersection between circles, and their centers is in fact a projection of a 3D cube on a certain plane, and there are three remaining sides of the cube which are not seen. Drawing them concludes the problem. It’s Quite cool. 🙂

I was unaware of who and when discovered this problem. It was discussed in at least a couple of books. The configuration appears quite naturally when considering the inversion in the incircle of a triangle. The problem also has an engaging informal solution.