AoPS - Problem

Circles

$S_1$ and

$S_2$ intersect at points

$M$ and

$N$ . Through point

$A$ of circle

$S_1$ , draw straight lines

$AM$ and

$AN$ intersecting

$S_2$ at points

$B$ and

$C$ , and through point

$D$ of circle

$S_2$ , draw straight lines

$DM$ and

$DN$ intersecting

$S_1$ at points

$E$ and

$F$ , and

$A$ ,

$E$ ,

$F$ lie along one side of line

$MN$ , and

$D$ ,

$B$ ,

$C$ lie on the other side (see figure). Prove that if

$AB = DE$ , then points

$A$ ,

$F$ ,

$C$ and

$D$ lie on the same circle, the position of the center of which does not depend on choosing points

$A$ and

$D$ .