The incircle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
of the acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
is tangent to its side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at a point
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
. Let
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
be an altitude of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, and let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the midpoint of the segment
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
. If
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
is the common point of the circle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
and the line
![KM](/media/m/0/1/c/01ca9badbb81b31fdf7b3e19f0f0c6c6.png)
(distinct from
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
), then prove that the incircle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
and the circumcircle of triangle
![BCN](/media/m/c/6/5/c658721c282eff4f5ce344d6a9d4de6c.png)
are tangent to each other at the point
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
.
%V0
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 $N$.