Consider a circular billiard table with a center O. A point (ball) located at the circumference is hit in an arbitrary direction and moves along an (infinite) trajectory reflecting from the circular edge. Suppose that the reflection law is such that for every three consecutive reflection points A, B, and C on the trajectory, angle ABO is equidistant to angle CBO. Prove that there is a circle inside the billiard such that the point's trajectory never crosses it.