Poncelet's Porism states that if a bicentric polygon exists on 2 circles then every point on the outer circle is the vertex of some bicentric polygon. (that is the "rotating" polygon above exists for every bicentric polygon)
The thick pink line represents the radical axis of the 2 circles with the limit circle being the pink dot on the X axis.
The configuration of the circles can be changed by dragging the pink line.
It has been proved that the diagonals of an even sided bicentric polygon intersect at the limit circle.
I could add that in the odd case it looks like the line through each vertex and the limit circle cuts the point where the inner circle and the opposite side touch.
A really good link
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