# Poncelet Porism in Ellipses

*Poncelet's porism* is an assertion that provided there is a polygon circumscribed around one conic and inscribed into another, there are infinitely many such polygons. For two circles, one nested in the other, a demonstration is available elsewhere.

The applet below illustrates Poncelet's porism in a more general situation of two ellipses, one inside the other.

Denote the bigger ellipse Γ and the smaller one γ. Choose point A on Γ and find a tangent from A to γ. Extend this tangent through the intersection with Γ at, say, B. For B, repeated the process of finding the tangent to γ and extending it to the intersection with Γ. Let the process go on. It is quite possible that one of the tangents will pass through the starting point A, thus closing a polygon inscribed in Γ and circumscribed around γ. Such a polygon is sometimes referred to as a *Poncelet's traverse*. The essence of Poncelet's porism is an assertion that if a traverse exists for one point A, a traverse will exist for any starting point A.

The porism is illustrated in the cases of a triangle and a quadrilateral. There are two ellipse: a blue within a red one. Both are defined by their foci and an extra point. On the red ellipse there is a solid point that is to serve as a vertex of a polygon - the meeting place of the first and the last tangents from one ellipse to the other. First adjust the ellipses as to close the polygon of the tangents and then drag the solid point.

### Conic Sections > Ellipse

- What Is Ellipse?

- Analog device simulation for drawing ellipses
- Angle Bisectors in Ellipse
- Angle Bisectors in Ellipse II
- Between Major and Minor Circles
- Brianchon in Ellipse
- Butterflies in Ellipse
- Concyclic Points of Two Ellipses with Orthogonal Axes
- Conic in Hexagon
- Conjugate Diameters in Ellipse
- Dynamic construction of ellipse and other curves
- Ellipse Between Two Circles
- Ellipse in Arbelos
- Ellipse Touching Sides of Triangle at Midpoints
- Euclidean Construction of Center of Ellipse
- Euclidean Construction of Tangent to Ellipse
- Focal Definition of Ellipse
- Focus and Directrix of Ellipse
- From Foci to a Tangent in Ellipse
- Gergonne in Ellipse
- Pascal in Ellipse
- La Hire's Theorem in Ellipse
- Maximum Perimeter Property of the Incircle
- Optical Property of Ellipse
- Parallel Chords in Ellipse
- Poncelet Porism in Ellipses
- Reflections in Ellipse
- Three Squares and Two Ellipses
- Three Tangents, Three Chords in Ellipse
- Van Schooten's Locus Problem
- Two Circles, Ellipse, and Parallel Lines

#### Poncelet Porism

- Poncelet's Porism
- Extouch Triangle in Poncelet Porism
- Fuss' Theorem
- Intouch Triangle in Poncelet Porism
- Euler's Formula and Poncelet Porism
- Poncelet Porism in Ellipses

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