2002 IMO Shortlist S06

The incircle $\Omega$ of the acute-angled triangle $ABC$ is tangent to its side $BC$ at a point $K$ . Let $AD$ be an altitude of triangle $ABC$ , and let $M$ be the midpoint of the segment $AD$ . If $N$ is the common point of the circle $\Omega$ and the line $KM$ (distinct from $K$ ), then prove that the incircle $\Omega$ and the circumcircle of triangle $BCN$ are tangent to each other at the point $N$ .

锐角三角形 $ABC$ 的内切圆 $\Omega$ 与其边 $BC$ 相切于点 $K$ 。令$AD$ 为三角形$ABC$ 的高，并令$M$ 为线段$AD$ 的中点。如果$N$是圆$\Omega$和线$KM$的公共点(与$K$不同)，则证明内切圆$\Omega$和三角形$BCN$的外接圆在点$N$处相切。