Let
be a fixed convex quadrilateral with
and
not parallel with
. Let two variable points
and
lie of the sides
and
, respectively and satisfy
. The lines
and
meet at
, the lines
and
meet at
, the lines
and
meet at
.
Prove that the circumcircles of the triangles
, as
and
vary, have a common point other than
.
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![BC=DA](/media/m/0/f/2/0f28e5dcf9fcc288cdec9e638f37a3b2.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![DA](/media/m/a/0/8/a081cf3dbb7eaedd62ea487a4cd46956.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![DA](/media/m/a/0/8/a081cf3dbb7eaedd62ea487a4cd46956.png)
![BE=DF](/media/m/1/0/6/106261aa9a7a3478d1de64e4dcc19c94.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.png)
Prove that the circumcircles of the triangles
![PQR](/media/m/e/d/5/ed5efbddbf500aeff476d02507a9f80a.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)