Question 20
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ with centre on the segment $AB$ is tangent to the equal sides $AC$ and $BC$. Consider a line tangent to $\Gamma$ that cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The lines perpendicular to $AC$ and $BC$ drawn through $D$ and $E$, respectively, meet at $P$. Let $Q$ be the foot of the perpendicular to the line $AB$ through $P$. Prove that
$$
\frac{PQ}{CP} = \frac{1}{2} \frac{AB}{AC}
$$
$$
\frac{PQ}{CP} = \frac{1}{2} \frac{AB}{AC}
$$