Question 44
In a convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and
$$
AB + DE = BC + EF = CD + FA.
$$
Denote the midpoints of sides $AB$, $BC$, $DE$, $EF$ by $A_1, B_1, D_1, E_1$, respectively.
Prove that $\angle D_1OE_1 = \frac{1}{2}\angle DEF$, where $O$ is the point of intersection of segments $A_1D_1$ and $B_1E_1$.
Supplemental figure:
$$
AB + DE = BC + EF = CD + FA.
$$
Denote the midpoints of sides $AB$, $BC$, $DE$, $EF$ by $A_1, B_1, D_1, E_1$, respectively.
Prove that $\angle D_1OE_1 = \frac{1}{2}\angle DEF$, where $O$ is the point of intersection of segments $A_1D_1$ and $B_1E_1$.
Supplemental figure: