Poncelet's porism

Poncelet's porism is a geometric principle stating that if there exists one closed polygon inscribed in a circle and simultaneously tangent to an inner circle, then there are infinitely many such polygons. Moreover, starting from any point on the outer circle, you can construct a polygon with the same number of sides, all touching the inner circle, by repeatedly drawing tangent lines. This phenomenon highlights a remarkable stability: once one such polygon exists, the process of creating similar polygons can be continued indefinitely, underscoring a special relationship between the two circles and the inscribed polygon.