Answer
Consider drawing a radius to each end of the common chord, thus creating two triangles. These two triangles share a side, which is congruent to itself by the identity property, and they also both have congruent right angles. Since the distance between the center of a circle and the end of a chord is the radius of one of the circles, these two circles have another identical side. By the Pythagorean theorem, this means that two sides divided by the original line are congruent. Thus, by the definition of a bisector, it follows that the original line is the perpendicular bisector of the common chord.
