Let $ABCC_1B_1A_1$ be a convex hexagon such that $AB=BC$, and suppose that the line segments $AA_1, BB_1$, and $CC_1$ have the same perpendicular bisector. Let the diagonals $AC_1$ and $A_1C$ meet at $D$, and denote by $\omega$ the circle $ABC$. Let $\omega$ intersect the circle $A_1BC_1$ again at $E \neq B$. Prove that the lines $BB_1$ and $DE$ intersect on $\omega$.
Školjka 
be a convex hexagon such that 
, and suppose that the line segments 
, and 
have the same perpendicular bisector. Let the diagonals 
and 
meet at 
, and denote by 
the circle 
. Let 
again at 
. Prove that the lines 
and 
intersect on