Poncelet's General Theorem

The theorem as stated in Komal a Hungarian Math Magazine
: Let e be a circle of a non-intersecting pencil and let a1,a2,...,an be (not necessarily different) oriented circles in the interior of e that belong to the same pencil. Starting at an arbitrary point A0 of the circle e, the points A1,A2,...,An are constructed on the same circle, such that the lines A0A1, A1A2, ..., An-1An touch the circles a1,a2,..., an, respectively, in the appropriate direction. It may happen that at the end of the construction, we get back to the starting point, that is, An=A0. The theorem states that in that case, we will always get back to the starting point in the n-th step, whichever point of e we start from. We do not even need to take care to draw the tangents to the circles in a fixed order.

The discussion in Komal is very easy reading it is intended for secondary students. I recommend reading it via the link above.

The Applet is to help explore this theorem.

You can add new points to the cyclic polygon by clicking on the black circle. You can drag points to another location on the circle. If you mark the delete mode radio button and then click on a point it deletes the point.
Every time you do this the edges of the polygon are adjusted and a circle tangential to the edge is made. All cicles are members of the pencil generated by the black circle and the radix. Of course you can drag the radix.

Poncelet's Theorem implies that once you create tangential circles(that are pencil members) for each edge of a cyclic polygon you can do a “rotation” by

• rotating the first point about the center of the black circle

• then drawing a new tangent to the first circle (the tangent must be on the same side as the original)

• extending the tangent to intersect the outer circle and call this intersect the second point

• repeat the above using the next circle to generate the third and subsequent points

• you will end up at the start point and a form of rotation will have happened

You can show this by checking the “rotate” radio button.

The theorem also predicts that if you change the order that the circles are used (I call this the traverse order) you still wind up back at the start point. You can see this by pressing the “Change Circle Traverse Order” button.