Let be a cyclic quadrilateral. Let be the intersection of lines parallel to and passing through points and , respectively. The lines and intersect the circumcircle of again at and , respectively. Prove that points , , , and lie on a circle.
Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.
Source: Srednjoeuropska matematička olimpijada 2015, pojedinačno natjecanje, problem 3
Školjka 
be a cyclic quadrilateral. Let 
be the intersection of lines parallel to 
and 
passing through points 
and 
, respectively. The lines 
and 
intersect the circumcircle of 
again at 
and 
, respectively. Prove that points 
, 
,