Question 43
In the scalene triangle $ABC$, with $\angle BAC = 90^\circ$, the inscribed and circumscribed circles are considered. The tangent line at $A$ to the circumcircle cuts the line $BC$ at $M$. Let $S$ and $R$ be the points of tangency of the circle inscribed with the legs $AC$ and $AB$, respectively. The line $RS$ cuts the line $BC$ at $N$. The lines $AM$ and $SR$ intersect at $U$. Prove that triangle $UMN$ is isosceles.