BALTIC-WAY 1997 Problem 13

Five distinct points $A, B, C, D$ and $E$ lie on a line with

$$|A B|=|B C|=|C D|=|D E| \text {. }$$

The point $F$ lies outside the line. Let $G$ be the circumcentre of triangle $A D F$ and $H$ be the circumcentre of triangle $B E F$. Show that lines $G H$ and $F C$ are perpendicular.