IMO 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.