An elementary proof of Poncelet's theorem on bicentric polygons

We give a new proof of Poncelet's theorem on bicentric polygons, using a generalisation of the notion of an orthocentre for an $n$-gon. We indicate some properties of bicentric polygons and find generalisations of Euler's formula connecting the radii of the inscribed and circumscribed circles and the distance between their centres for convex $n$-gons with $n=4, 5, 6$, and also for a non-convex pentagon. In conclusion, we consider a construction of three related bicentric pentagons.
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