We are given a cyclic quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
with a point
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
on the diagonal
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
such that
![AD=AE](/media/m/b/7/8/b783baa7685810794199b44d2aedbed6.png)
and
![CB=CE](/media/m/e/a/8/ea88e3cdc575eb63b30977e589cc8320.png)
. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be the center of the circumcircle
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
of the triangle
![BDE](/media/m/5/2/e/52e30a04d484e9018ded169769e8f25f.png)
. The circle
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
intersects the line
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
in the points
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
. Prove that the lines
![FM](/media/m/5/7/d/57d8386a345f7f01c20faf11567ac3d7.png)
,
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
meet at one point.
%V0
We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point.