A Short Proof of a Ptolemy-Like Relation for an Even number of Points on a Circle Discovered by Jane McDougall

Marc Chamberland
Doron Zeilberger

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(Published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and also appeared under the shorte title "A short proof of McDougall's Circle Theorem" in the Amer. Math. Monthly v. 121(2014), 263-265)

Written: April 15, 2013.

          In Fond memory of Andrei Zelevinsky (1953-2013)

One of my favorite people was Andrei Zelevinsky, who sadly died last week, and one of Andrei's favorite theorems was Ptolemy's Theorem. The deep Gelfand-Kapranov-Zelevinsky theory may be viewed as a very long commentary on Ptolemy's theorem, whose fancy name is ``Plücker relations''.

Here we give a short proof of a Ptolemy-style result (that for all we know may be a special case of something in the Gelfand-Kapranov-Zelevinsky theory), first discovered and proved by Jane McDougall (see her message).

Added Oct. 2013: slightly expanded version will appear in the Amer. Math. Monthly

Acknowledgement: This is yet another instance of the power of Lagrange Interpolation, preached by Edi Gnang.

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

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