Mowaffaq Hajja, A very short and simple proof of ``the most elemetary theorem'' of euclidean geometry,
Forum Geometricorum, 6 (2006) 167--169.

Abstract: We give a very short and simple proof of the fact that if ABB' and AC'C are straight lines with BC and B'C' intersecting at D, then  AB + BD = AC' + C'D if and only if AB' + B'D = AC + CD. The ``only if" part is attributed to Urquhart, and is referred to by Dan Pedoe as ``the most elementary theorem of Euclidean geometry".

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