2006 OIM Problems/Problem 1

Problem

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 intersects the line $BC$ at $M$. Let $S$ and $R$ be the points of tangency of the circle inscribed with the lines $AC$ and $AB$, respectively. Line $RS$ intersects line $BC$ at $N$. Lines $AM$ and $SR$ meet at $U$. Show that the triangle $UMN$ is isosceles.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions