International Mathematical Olympiad 1960 Problem 7

An isosceles trapezoid with bases $a$ and $c$ and altitude $h$ is given.

(a) On the axis of symmetry of this trapezoid, find all points $P$ such that both legs of the trapezoid subtend right angles at $P$.

(b) Calculate the distance of $P$ from either base.

(c) Determine under what conditions such points $P$ actually exist. (Discuss various cases that might arise.)