Let
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
be a cyclic quadrilateral. Let
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
be variable points on the sides
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
and
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
, respectively, such that
![AE:EB=CF:FD](/media/m/a/d/2/ad2bcf22d985bd02acf991841ffb78a4.png)
. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be the point on the segment
![EF](/media/m/f/5/5/f5594d5ec47ea777267cf010e788fedd.png)
such that
![PE:PF=AB:CD](/media/m/f/d/1/fd19d7357064fb37e20590741e1e08f1.png)
. Prove that the ratio between the areas of triangles
![APD](/media/m/5/f/0/5f0d8f53b46f1d7515af208c6cd58339.png)
and
![BPC](/media/m/4/4/0/440b9609d991a008cee182246acbb9d6.png)
does not depend on the choice of
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
.
%V0
Let $ABCD$ be a cyclic quadrilateral. Let $E$ and $F$ be variable points on the sides $AB$ and $CD$, respectively, such that $AE:EB=CF:FD$. Let $P$ be the point on the segment $EF$ such that $PE:PF=AB:CD$. Prove that the ratio between the areas of triangles $APD$ and $BPC$ does not depend on the choice of $E$ and $F$.