[ Right triangle with an angle in ]

From the vertex of the right angle of triangle $ABC$, the median $CM$ is drawn. The circle inscribed in triangle $CAM$ touches $CM$ at its midpoint. Find the angle $BAC$.

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Denoting by $K, P$ and $H$ the points of tangency of the circle with the sides of triangle $C A M$ (see figure), we get: $A K=$ $A P, C K=C H=M H=M P$. Therefore, $A C=A M$. Moreover, $C M=0.5 A B=A M$.

Thus, triangle $C A M$ is equilateral, which means $\angle B A C=60^{\circ}$.

## Answer

$60^{\circ}$.