Two circles
![\Omega_{1}](/media/m/d/d/c/ddcc714ea0eb5673e37799e314dbf8bd.png)
and
![\Omega_{2}](/media/m/3/c/c/3cc19d2e9ac0b321a4a565a07b249f98.png)
touch internally the circle
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
in M and N and the center of
![\Omega_{2}](/media/m/3/c/c/3cc19d2e9ac0b321a4a565a07b249f98.png)
is on
![\Omega_{1}](/media/m/d/d/c/ddcc714ea0eb5673e37799e314dbf8bd.png)
. The common chord of the circles
![\Omega_{1}](/media/m/d/d/c/ddcc714ea0eb5673e37799e314dbf8bd.png)
and
![\Omega_{2}](/media/m/3/c/c/3cc19d2e9ac0b321a4a565a07b249f98.png)
intersects
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
in
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
.
![MA](/media/m/1/a/2/1a298617d983de609dc306755b66a265.png)
and
![MB](/media/m/5/d/c/5dcd44e421f0875ab7ce126cece0fe27.png)
intersects
![\Omega_{1}](/media/m/d/d/c/ddcc714ea0eb5673e37799e314dbf8bd.png)
in
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. Prove that
![\Omega_{2}](/media/m/3/c/c/3cc19d2e9ac0b321a4a565a07b249f98.png)
is tangent to
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
.
%V0
Two circles $\Omega_{1}$ and $\Omega_{2}$ touch internally the circle $\Omega$ in M and N and the center of $\Omega_{2}$ is on $\Omega_{1}$. The common chord of the circles $\Omega_{1}$ and $\Omega_{2}$ intersects $\Omega$ in $A$ and $B$. $MA$ and $MB$ intersects $\Omega_{1}$ in $C$ and $D$. Prove that $\Omega_{2}$ is tangent to $CD$.