Steiner's Porism

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The diagram below shows an example of Steiner's Porism. We begin with two concentric circles, and draw the circles tangent to both (there are seven in the diagram below). Now apply an inversion through the red circle. The result is the pair of circles on the right, again admitting a Steiner configuration of seven circles.

You can move the centre \(O\) and the point \(P\) on the red circle to change the circle of inversion, to see how this affects the diagram. Steiner's porism remains true since inversion preserves tangency of circles.