7. A point is on the perpendicular bisector of a line segment if and only if it lies the same distance from the two endpoints.

There are two things that need to be proved here. The first is that if a point is on the perpendicular bisector of a line segment, then it is equidistant from the two endpoints of the segment.

Given: AC = CB, DC is perpendicular to AB.

To prove: AD = BD

This follows directly from Theorem 5 of this section.

The other thing that needs to be proved is that if D is equidistant from A and B, then it is on the perpendicular bisector of AB.

This follows directly from Theorem 3 of this section.