Poncelet's theorem for the circle
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The diagram above contains two circles. Given an initial point A on the outer circle, draw a tangent from A to the inner circle, and then continue until it intersects the outer circle again at a new point B. The resulting line is a chord of the outer circle which is tangent to the inner circle. Then repeat the process and draw a new chord from B which is tangent to the inner circle and intersects the outer circle again at a new point C. Repeating this process over and over produces a diagram like the one above, which contains eight of these tangential chords.
If the chords ever return to the point A, then the process of constructing the chords repeats itself, and the resulting figure is called a Poncelet Polygon.
Poncelet's theorem says that if a Poncelet polygon exists then it is independent of the initial point A. You can verify this by moving A around the circle or by clicking the button in the bottom left corner to animate the picture. Click and drag the points \(O_1\) and \(P_1\) to change the centre and radius of the outer circle, and move the points \(O_2\) and \(P_2\) to change the centre and radius of the inner circle. You can see that if the two circles are positioned so that the chords close up to form a Poncelet polygon, then they will do so for every choice of the initial point A on the outer circle. Conversely, if they do not close up to form a Poncelet polygon, then the same will be true for every initial point A.
After experimenting with the picture above, you can see that it is difficult to get the two circles exactly right so that the chords form a perfect Poncelet polygon. If you want to construct perfect Poncelet triangles and quadrilaterals, then it is possible to use a ruler and compass to find the exact positions of the two circles. The two links below will show you how to do this.
Constructing Poncelet Triangles
Constructing Poncelet Quadrilaterals
