IMO 1979 Problem 3

I "solved" another International Mathematical Olympiads problem:

There are two circles in the plane. Let a point A be one of the points of intersection of these circles.
Two points begin moving simultaneously with constant speeds from the point A, each point along its own circle.
The two points return to the point A at the same time.
Prove that there is a point P in the plane such that at every moment of time 
the distance from the point P to the moving points are equal.

The demo below draws all the lines that satisfy the condition while the points are moving, if there is such a point P then our lines must intersect in that point.

r (R1/R2)

q [R1 - R2, R1 + R2]

Move

Inverse direction

The source code can be found here: http://github.com/ex/js
And the stand alone demo can be viewed here: http://ex.github.io/js/src/imo_1979_3/