Poncelet's closure theorem

Statement

Let $C$ and $D$ be ellipses in the Euclidean plane with $D$ in the interior of $C$ . For $P_0 \in C$ , build a sequence $P_0, P_1, P_2, \ldots$ on $C$ as follows. From $P_k$ there are two tangent lines to $D$ . For $k \ge 1$ , take the one other than the line $P_{k-1} P_k$ ; for $k = 0$ , take either one. Let $P_{k+1}$ be the second point where this tangent line meets $C$ . Suppose that for one starting point $P_0$ the sequence satisfies $P_n = P_0$ for some $n \ge 3$ . Then $P_n = P_0$ holds for every starting point $P_0 \in C$ .

Remarks

The polygon $P_0 P_1 \cdots P_{n-1}$ is inscribed in $C$ and circumscribed about $D$ . A modern proof considers the curve of pairs $(P, \ell)$ with $P \in C$ and $\ell$ a tangent line of $D$ through $P$ . Over the complex numbers this curve is an elliptic curve, and one step of the construction is a translation on it. A translation that has a point of period $n$ has every point of period $n$ . Example: for two concentric circles with radii $R$ and $R/2$ , every inscribed equilateral triangle is circumscribed about the inner circle, so every start closes after $3$ steps. For a triangle and two circles, closure happens exactly when Euler's theorem $d^2 = R(R - 2r)$ holds between the radii and the distance $d$ of the centres.

Going deeper

For a student

Poncelet's theorem is a porism: the closing property is all or nothing. It holds more generally for two smooth conics in general position in the complex projective plane. Its proof through elliptic curves is a standard illustration of how the group law of a cubic curve solves a problem of elementary geometry.

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Ellipse concept
Limit point of a set definition
Limit of a function definition
Derivative definition
Tangent line concept
Poncelet's closure theorem statement

Tag	`XWYXSRM5YS`
URI	`m2://XWYXSRM5YS`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04, 14H52
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About	Ellipse
Uses	Ellipse, Tangent line
Same as	Wikidata Q1785610 Wikipedia Poncelet's closure theorem