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We have a circle $\Gamma$ and two points $A$, $A'$ on this circle. We have a line $\Delta$ and a point $P$ on this line: $P \in \Delta$.

Do you know a way to construct points $M \in \Gamma$ such that $a=MA \cap \Delta$ et $a'=MA' \cap \Delta$ and $P$ is the middle of $[aa']$. I'm trying to use central symmetries but I'm still searching ! Many thanks for any idea !

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When observing reflection on line $ \Delta$ across $P$: $a\mapsto a'$ which induces projective transformation $Aa\mapsto A'a'$ from a pencil of line through $A$ to a pencil of lines trhough $A'$ we see that intersection point $M=Aa\cap A'a'$ describes some conic (since $AA'$ does not map to it self). So $M$ is intersection point of this conic and circle.

Playing in Geogebra I got some hyperbola. So I doubt there is a nice elementary solution.

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