Let C and D be distinct points on the semicircle with diameter AB.
Let rays AC and BD intersect at point E and let F be the foot of the
the perpendicular from E to AB. Let P be the intersection of the
perpendicular bisector of AB and line CD and Q a point on CD,
distinct from P.

Prove that FQ ⊥ CD if and only if ∠PAQ = ∠PBQ.