An if-and-only-if theorem has the form '"p" if and only if "q"', where "p" and "q" are statements. Such a theorem is equivalent to asserting two statements are true:
(1) 'if "p" then "q"' and also
(2) 'if "q" then "p"'.
So the proof of the theorem will have TWO proofs, one for (1) and the other for (2).
Theorem 1: A point P is on the perpendicular bisector of line segment AB if and only if PA = PB.
(see B&B, p. 88, Principle 10)
This is the same as saying these two statements are true:
(1) If P is on the perpendicular bisector of line segment AB, then PA = PB.
(2) For two points A and B, if PA = PB, then P is on the perpendicular bisector of segment AB.
Notice that these two statements are the ones proved in Assignment Problem 1.3. Thus Problem 1.3 (b) and (c) could have been restated as one statement to prove, the Theorem above.
There is another, more visual way of restating the theorem above.
Theorem 2: The locus of points equidistant from two points A and B is the perpendicular bisector of segment AB.
(see B&B, p. 250, Locus Theorem 4)