Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a convex quadrilateral and let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
be points in
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
such that
![PQDA](/media/m/6/2/b/62bc01a87eb27d086281b5805edf88c3.png)
and
![QPBC](/media/m/f/e/b/feb9cfb76e980a42902875bfb06f8aec.png)
are cyclic quadrilaterals. Suppose that there exists a point
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
on the line segment
![PQ](/media/m/f/2/f/f2f65ec376294df7eca22d2c1a189747.png)
such that
![\angle PAE = \angle QDE](/media/m/e/7/3/e73bdc53d6897200e209dfb256bb9a5c.png)
and
![\angle PBE = \angle QCE](/media/m/6/5/6/6565ab1cc8806a8079b97d418eedd6ce.png)
. Show that the quadrilateral
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
is cyclic.
Proposed by John Cuya, Peru
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Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be points in $ABCD$ such that $PQDA$ and $QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $E$ on the line segment $PQ$ such that $\angle PAE = \angle QDE$ and $\angle PBE = \angle QCE$. Show that the quadrilateral $ABCD$ is cyclic.
Proposed by John Cuya, Peru