KöMaL 2011 B4383

ABCD is a convex quadrilateral, such that $AB$>$BC$>$CD$>$DA$. The inscribed circles of the triangles ABD and BCD touch the diagonal $BD$ of the quadrilateral at the points $E$and $F$, respectively. Show that $EF$=$GH$. (Suggested by Sz. Miklós, Herceghalom)