Question 40
An altitude from $A$ to the triangle $ABC$ intersects the side $BC$ in $D$. A circle touches $BC$ in $D$, intersects $AB$ in $M$ and $N$ and intersects $AC$ in $P$ and $Q$. Prove that
$$
\frac{AM + AN}{AC} = \frac{AP + AQ}{AB}.
$$
$$
\frac{AM + AN}{AC} = \frac{AP + AQ}{AB}.
$$