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Given a point on a torus there exist exactly 4 circumferences that lie on the surface of the torus passing through that point. The two non-obvious ones are called Villarceau circles.

Wikipedia gives a proof of their existence with analytic geometry. I'm interested in a proof with synthetic geometry because of this anecdote by Emilio Segrè:

On a different occasion, while I was waiting to be called to an oral examination, Majorana gave me a synthetic proof for the existence of Villarceau's circles on a torus. I did not fully understand it, but memorized it on the spot. As I entered the examination room, Professor Pittarelli asked me, as was his wont, whether I had prepared a special topic. "Yes, on Villarceau's circles," I said, and I proceeded immediately to repeat Majorana's words before I forgot them. The professor was impressed and congratulated me on such an elegant proof, which was new to him.

(From here, emphasis mine)

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I could say that, although it doesn't answer this question, maybe this is a good occasion to mention that Wikipedia's List of circle topics is a magnificent thing, the like of which might never have existed if Wikipedia, itself an unprecedented achievement, had not existed. I could say that, but since I'm the initial author of that article, I won't, but instead will merely say that if you're curious about this kind of geometry, you might find it of some value. –  Michael Hardy Mar 23 '12 at 18:29
    
+1 for the question. Often synthetic proofs shed intuitive light not found in analytic proofs (and probably vice-versa as well). –  Michael Hardy Mar 23 '12 at 18:30

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This proof can be found in Anonymous - Section du tore par un plan tangent à cette surface et passant par son centre in Nouvelle Annales de Mathématiques 1.18 (1859), pp. 258-261, which is available here.

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