Poncelet-type problems, an elementary approach
This article presents a survey on three geometric closure theorems, the well-known Poncelet theorem (two-circle version) and the so-called Zig-zag and Ponzag theorems. In Euclidean space \(E^3\) let both the objects \(K\) and \(L\) be either a circle or a straight line. Let be also given a fixed length \(r>0\) and a pair of points \(A_1\in K\), \(B_1\in L\) having this distance \(r\) from each other; it is assumed that neither \(K\) nor \(L\) has a point from which all the points of the other object are at distance \(r\). Now construct a polygon \(A_1B_1A_2 B_2\dots\) according to the following recursion (``Zig-zag process''): For \(n\in \mathbb{N}\) let \(A_{n+1}\) \((B_{n+1})\) be that point on \(K\) \((L)\) which is at distance \(r\) from \(B_n\) \((A_{n+1}\) -- not \(A_n\), as quoted by mistake on page 46, line 15 of the article!) and which is different from \(A_n\) \((B_n)\), respectively; if no such point exists than put \(A_{n+1}= A_n\) \((B_{n+1} = B_n)\), respectively. Then the following, basic ``Zig-zag theorem'' (Section 1) holds: If a Zig-zag process, starting from some given segment \(A_1B_1\) of length \(r\) \((A_1\in K,\;B_1\in L)\) is of periodicity \(n\) (i.e. \(B_{n+1} =B_1)\), then any other initial segment of the same length \(r\) gives rise to a Zig-zag process of the same periodicity \(n\). In a similar way, the search for polygons inscribed in a circle \(K\) whose midpoints are also on some circle \(L\) (Section 2) and for polygons which have both inscribed and circumscribed circles \(L,K\) (Section 3) leads to periodic processes of of the so-called Ponzag and the (well-known) Poncelet type, respectively. The corresponding Ponzag theorem (shortly: ``If there exists an \(n\)-gon whose vertices lie on the circle \(K\) and the midpoints of the sides lie on another circle -- or a straight line \(L\)-, then an infinite number of such \(n\)-gons exist'') and the Poncelet theorem (two-circle version) are formulated. In the final Section 4 the equivalence of the three theorems Zig-zag/Ponzag/Poncelet is demonstrated.
polygons having both inscribed and circumscribed circles
scientific article; zbMATH DE number 1475280