Question 104
In an acute scalene triangle $\triangle ABC$ points $P, Q$ lie on the smaller arcs $\widehat{AB}$ and $\widehat{AC}$, respectively, and are the intersection of the midline parallel to $BC$ with the circumcircle of $\triangle ABC$. Points $X$ and $Y$ are the intersection of the perpendicular bisectors of segments $AB$ and $AC$, respectively, with the tangent of $\odot(ABC)$ at point $A$. Define point $T \neq A$ as the intersection of $\odot(PXA)$ and $\odot(QYA)$. If $AD$ is an altitude in the triangle $\triangle ABC$, prove that
$$ \frac{|TX|}{|TY|} = \frac{|DB|}{|DC|} $$
$$ \frac{|TX|}{|TY|} = \frac{|DB|}{|DC|} $$