EGMO 2020 Problem 3

Let $ABCDEF$ be a convex hexagon such that $\angle A = \angle C = \angle E$ and $\angle B = \angle D = \angle F$ and the (interior) angle bisectors of $\angle A, ~\angle C,$ and $\angle E$ are concurrent.

Prove that the (interior) angle bisectors of $\angle B, ~\angle D,$ and $\angle F$ must also be concurrent.

Note that $\angle A = \angle FAB$ . The other interior angles of the hexagon are similarly described.