Let $\mathscr{C}$ be a circle of centre $O$, $\mathscr{P}:= [a_1, \ldots, a_n]$ and $\mathscr{Q}:= [b_1, \ldots, b_m]$ two convex polygons, inscribed in $\mathscr{C}$ such that:
- $O$ is inside $\mathscr{P}$ and $\mathscr{Q}$.
- The shortest side of $\mathscr{P}$ is bigger than the longest side of $\mathscr{Q}$.
Then the perimeter de $\mathscr{P}$ is smaller than the perimeter de $\mathscr{Q}$.
I have a proof using trigonometry and a convex (or concave) inequality.
However I would be interested in a proof at middle school level, not using the notion of function. Any idea?