Balls, wrong image. One second.

Proof. Let q be a point on the other side of L from p. Let \alpha be the circle through q with center p. Let a and b be the intersection points of L and \alpha. By Proposition I.10, let d bisect the segment from a to b. Let M be the line through p and d. By Proposition I.8, \angle pda = \angle pdb, hence L and M are perpendicular.