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Let $A$, $B$, $C$ and $D$ be four circles passing through a common point

Each pair of circles intersects in another point, thus giving rise to 6 new points: $p_{AB}$, $p_{AC}$, $p_{AD}$, $p_{BC}$, $p_{BD}$, $p_{CD}$

For each triplet of points (e.g. $p_{AB}$, $p_{AC}$, $p_{BC}$) coming from three of the four initial circles (e.g. $A$, $B$, $C$), we introduce the circle passing through them (e.g. $\Gamma_{ABC}$)

Then the four circles $\Gamma_{ABC}$, $\Gamma_{ABD}$, $\Gamma_{ACD}$ and $\Gamma_{BCD}$ all pass through a common point