Convex Bodies with Equipotential Circles

Given a convex body K subset of R2 we say that a circle B subset of intK is an equipotential circle if a variable chord AB of K tangent to B at P has the product |AP|<middle dot>|PB| constant. The main result of this article is the following: Let K subset of R2 be a convex body that has an interior equipotential circle B centered at O. Then K has center of symmetry at O, moreover, if no chord of K tangent to B subtends an angle pi/2 from O, then K is a disk.