Light from C reflects in the tangent to the circle at point B.

Observe the trace as B rotates around the circle.  While the individual rays move, the phantom curve – the caustic – does not.

The caustic is the mathematical envelope of the reflected rays, which is to say that each reflected ray is tangential to the envelope curve.

At any moment, each ray is moving with a combination of rotation and translation.  Each point on the ray is undergoing a translation.  Points on the ray which lie on the caustic are moving along teh length of the ray, and hence their motion is not visible.  Which is why the caustic looks stationary while everything else moves.