bicyclic polygons
If you extend the opposite sides of an even sided bicyclic polygon they intersect in a line.
The line is a tangent to the circle orthogonal to the inner and outer circles of the bicyclic polygon.
You can alter the distance between the center of the circles by clicking the buttons"<" and ">" next to the "center distance" label.

Pascal's Theorem states that the opposite intersects of a circular hexagon are linear.
The projection.
Imagine a light above the center of the outer circle(towards the viewer if on screen)This light projects the image onto a surface that is parallel to the plane containing the light and the (red) intersect line.
The effect of this is to move the intersect line to infinity and make the opposite intersect lines into parallels. To prevent the image from being moved off screen I have stretched,resized and moved it so that the outer circle is the same size and position as the original outer circle.

A great link from Hungary
Eric W. Weisstein. "Poncelet's Porism." From MathWorld--A Wolfram Web Resource.
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