Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a cyclic quadrilateral whose diagonals
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
meet at
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
. The extensions of the sides
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
and
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
beyond
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
meet at
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
. Let
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
be the point such that
![ECGD](/media/m/5/3/c/53c073a613a37c08548f818a828194c3.png)
is a parallelogram, and let
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
be the image of
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
under reflection in
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
. Prove that
![D,H,F,G](/media/m/f/9/2/f92ae2a8c32bdbd7761435209413ffed.png)
are concyclic.
%V0
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.