Question 68
Given two points $A$ and $B$, take $C$ on the perpendicular bisector of $AB$. Define the sequence $C_{1}, C_{2}, C_{3}, \ldots$ as follows. $C_{1}=C$. If $C_{n}$ is not on $AB$, then $C_{n+1}$ is the circumcenter of the triangle $ABC_{n}$. If $C_{n}$ lies on $AB$, then $C_{n+1}$ is not defined and the sequence terminates. Find all points $C$ such that the sequence is periodic from some point on.