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

And the stand alone demo can be viewed here: http://ex.github.io/js/src/imo_1979_3/