IGO 2018 A5

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$ . Another circle passing through $C,D$ is tangent to $AB$ at point $F$ . Point $G$ is the intersection point of $AE,DF$ , and point $H$ is the intersection point of $BE$ , $CF$ . Prove that the incenters of triangles $AGF$ , $BHF$ , $CHE$ , $DGE$ lie on a circle.

Proposed by Le Viet An (Vietnam)