There are many ways to prove it, and the geometric method is relatively simple. Just give a brief description:

Let the center of the circle be O, the radius be R, the fixed point be P, and the midpoint of PO be O'. For any point A on the circle O, the midpoint of PA is A', which obviously means that O'A'= OA/2 = R/2 = a constant value, that is, the trajectory is a circle with O' as the center.