Let circles A and B have radii a and b respectively.
Letting M be the origin and using polar coordinates
the circles have the following equations:
Circle A: r = 2a cos(theta)
Circle B: r = 2b cos(theta)
where theta is the angle between r and the line
determined by the centers of the circles.
Let Q' be the intersection of MQ and circle A.
Let R' be the intersection of MR and circle A.
MQ' 2a cos(theta_q) a 2a cos(theta_r) MR'
----- = ----------------- = --- = ----------------- = -----
MQ 2b cos(theta_q) b 2b cos(theta_r) MR
==> triangles Q'MR' and QMR are similar
==> Q'R' and QR are parallel
==> AP and Q'R' are perpendicular
==> <PAQ' = <PAR'
==> <PMQ = <PMQ' = (<PAQ')/2 = (<PAR')/2 = <PMR' = <PMR