Question 100
Given a right-angled triangle $ABC$ with $\angle C = 90^\circ$.
Triangle $AMN$ equal to $ABC$ is constructed on the hypotenuse of $ABC$, $\angle ANM = 90^\circ$, $AN = BC$ (see the fig.). The incircle $\Gamma_1$ of triangle $AMN$ touches the hypotenuse $AM$ at point $P$, and the incircle $\Gamma_2$ of triangle $ABC$ touches the side $BC$ at point $Q$.
Prove that the segment $PQ$, the hypotenuse $AB$, and the segment connecting the centers of $\Gamma_1$ and $\Gamma_2$ are concurrent.
Triangle $AMN$ equal to $ABC$ is constructed on the hypotenuse of $ABC$, $\angle ANM = 90^\circ$, $AN = BC$ (see the fig.). The incircle $\Gamma_1$ of triangle $AMN$ touches the hypotenuse $AM$ at point $P$, and the incircle $\Gamma_2$ of triangle $ABC$ touches the side $BC$ at point $Q$.
Prove that the segment $PQ$, the hypotenuse $AB$, and the segment connecting the centers of $\Gamma_1$ and $\Gamma_2$ are concurrent.