Question 3
Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^\circ$. Point $H$ is the foot of the perpendicular from $A$ to $\overline{BD}$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside triangle $SCT$ and $$
\angle CHS - \angle CSB = 90^{\circ},
\quad \angle THC - \angle DTC = 90^{\circ}.
$$ Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.
\angle CHS - \angle CSB = 90^{\circ},
\quad \angle THC - \angle DTC = 90^{\circ}.
$$ Prove that line $BD$ is tangent to the circumcircle of triangle $TSH$.