Given trapezoid
with parallel sides
and
, assume that there exist points
on line
outside segment
, and
inside segment
such that
. Denote by
the point of intersection of
and
, and by
the point of intersection of
and
. Let
be the midpoint of segment
, assume it does not lie on line
. Prove that
belongs to the circumcircle of
if and only if
belongs to the circumcircle of
.
Proposed by Charles Leytem, Luxembourg
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![\angle DAE = \angle CBF](/media/m/b/4/3/b43daad1cfff26f0beb808517a29ae0e.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
![ABK](/media/m/9/a/e/9ae8d3effb3c35999014d27caccf8949.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![CDJ](/media/m/3/5/0/350e5950b1b097755cb8542db41d90bc.png)
Proposed by Charles Leytem, Luxembourg