Let $ABC$ be a triangle. Let the external bisector of angle $A$ meet the circumcircle of triangle $ABC$ again at $M \neq A$. A circle with centre $M$ and radius $MB$ meets the internal bisector of angle $A$ at points $P$ and $Q$. Determine the length of $PQ$ in terms of the lengths of $AB$ and $AC$.
Could anyone please provide a solution? I cannot seem to make any significant progress in the question.
Edit: Here is the original project that I created in Geogebra. Hope it makes the diagram clearer.