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Journal for Geometry and Graphics 08 (2004), No. 1, 017--022
Copyright Heldermann Verlag 2004

An Elementary Proof of "the Most Elementary Theorem" of Euclidean Geometry

Mowaffaq Hajja
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

We give a fairly elementary proof of the fact that if ABB' and AC'C are triples of collinear points with the lines BC and B'C' intersecting at D, then d(AB) + d(BD) = d(AC') + d(C'D) if and only if d(AB') + d(B'D) = d(AC) + d(CD), where d(XY) denotes the length of the line segment joining X and Y. The "only if" part of this theorem is attributed to Urquhart, and referred to by Dan Pedoe as the most elementary theorem of Euclidean Geometry. We also give a simple proof of a variant of Urquhart's theorem that was discovered by Pedoe.

Keywords: Geometry of quadrangles, Urquhart's theorem.

MSC: 51M04

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