Given $2n+1$ fixed points: $A\_1, A\_2,....,A\_{2n+1}$ and point $P$. Let $B\_1$ is the reflection of $P$ in $A\_1$, $B\_2$ is the reflection of $B\_1$ in $A\_2$,...., $B\_{2n+1}$ is the reflection of $B\_{2n}$ in $A\_{2n+1}$. 



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> > **My question:** Could You show that midpoints of $PB\_{2n+1}$ is fixed point when $P$ is moved.
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From a point and continuing reflection in $2n+1$ points then midpoint of the end point and the first point is fixed
The reflection of a point $p$ in a point $u$ is given (if we identify points with vectors) by $2u-p$. So the composition of two such reflections, say about points $u$ and $v$, is given by the translation $p\mapsto2v-(2u-p)=p+2(v-u)$. By induction, the composition of reflections in $u\_1,v\_1,u\_2,v\_2,\dots u\_n,v\_n$ is the translation $p\mapsto p+2(V-U)$ where $V$ and $U$ are the sums of the $v\_i$'s and $u\_i$'s, respectively. Composing with one last reflection in a point $w$, we get $p\mapsto 2w-p-2(V-U)$. The midpoint between the initial point $p$ and the result of all these reflections is therefore $w-(V-U)=w+U-V$, which is independent of $p$.


