If I'm not mistaken, we can prove that $ACE$ must be an equilateral triangle. Then the result follow by consider the center of the equilateral triangle as center of circle that will inscribed in the hexagon.

Or prove that angle bisectors of $ABCDEF$ concur.

let $O$be the circumceter of triangle $FDB$ we know that$AO$ and $CO$ and $EO$ are bisector of angles and they councr at $O$ the show that $ ABCDEF$ciclyc wich is solve the problem.

if you dont know why this hexagon is cyclic read this hint: show that $AFDC$,... cyclic . (use similarity)