MathNet 2012 Question 24

$ABCD$ is a cyclic quadrilateral. Let the circle $\Gamma_1$ pass through points $A$ and $B$ and touch $CD$ at point $E$; let the circle $\Gamma_2$ pass through points $B$ and $C$ and touch $DA$ at point $F$; let the circle $\Gamma_3$ pass through points $C$ and $D$, and touch $AB$ at point $G$; let at last the circle $\Gamma_4$ pass through points $D$ and $A$ and touch $BC$ at point $H$.
Prove that $EG \perp FH$.
(A. Voidelevich)