## Balkan Mathematical Olympiad 2012 Problem 1

Let , and be points lying on a circle with centre . Assume that . Let be the point of intersection of the line with the line perpendicular to at . Let be the line through which is perpendicular to . Let be the point of intersection of with the line , and let be the point of intersection of with that lies between and . Prove that the circumcircles of triangles and are tangent at .

*Balkan Mathematical Olympiad 2012 Problem 1*

**Solution: **Note that the circles are tangent if , and this condition is equivalent to . Since is the orthocenter of the triangle formed by it follows that , and the desired condition is equivalent to , which is true because is cyclic.

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