Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle with
![AB<AC](/media/m/c/3/6/c362c4e3344de08a4a6dac4af97efa8d.png)
and let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be the centre of its circumcircle
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
be a point on the line segment
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
such that
![\angle BAD = \angle CAO](/media/m/5/3/4/5342fa232d4c8859754f0d7e00948e6d.png)
. Let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
be the second point of intersection of
![\omega](/media/m/d/5/8/d58b95547061c22be95770ed7010f287.png)
and the line
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
. If
![M, N](/media/m/7/1/1/7119fd5b5b7ecea73d6b1400ec6abdb7.png)
and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
are the midpoints of the line segments
![BE, OD](/media/m/1/6/4/16462fd003b60d25f78abb942b8ea841.png)
and
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
, respectively, show that the points
![M, N](/media/m/7/1/1/7119fd5b5b7ecea73d6b1400ec6abdb7.png)
and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
are collinear.
%V0
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M, N$ and $P$ are the midpoints of the line segments $BE, OD$ and $AC$, respectively, show that the points $M, N$ and $P$ are collinear.