I could have labelled this Poncelet’s closure theorem, but have gone for a more populist approach.
Poncelet was a highly original mathematician. In particular he was French, 19th century, and a geometer who specialised in the projective variety. He has other strings to his bow – googling his name will give you the option of heading for the Poncelet (water) wheel.
And a porism is a special category of mathematical result.
Imagine two concentric circles. If they’re in the right proportion you could draw a triangle (or square) in the gap between the circles so that the triangle (or square) touches each circle three (or four) times.
Another way of thinking about such a drawing is to start with two circles, pick a point on the outer circle, draw a line that touches the inner circle and continues ’til it meets the outer circle – and to repeat until… Well, what might happen ? Either you get back to where you started – the case represented by the triangle or square – or you go on for ever. (This dual outcome is what constitutes a porism).
Two concentric circles turn out to be the simplest case. What if the circles aren’t concentric ? What if the circles become ellipses, parabolas, hyperbolas – which in projective geometry are equivalent ? You don’t need me to tell you that this extends the variety and complexity considerably. Poncelet sorted it out in around 1820.
I had intended to embed an animated diagram (the pretty picture) right here, showing six points and two ellipses – but I’m not clever enough. Either that or the image needs access to the necessary software (geogebra) – so you’ll need to follow this link instead.