
Journal for Geometry and Graphics 08 (2004), No. 1, 017022 Copyright Heldermann Verlag 2004 An Elementary Proof of "the Most Elementary Theorem" of Euclidean Geometry Mowaffaq Hajja Dept. of Mathematics, Yarmouk University, Irbid, Jordan mhajja@yu.edu.jo 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 [ Fulltextpdf (153 KB)] for subscribers only. 