Geodesic
An Elementary Proof of "the Most Elementary Theorem" of Euclidean Geometry
Journal for geometry and graphics, Tome 8 (2004) no. 1, pp. 17-22 Cet article a éte moissonné depuis la source Heldermann Verlag

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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.
Classification : 51M04
Mots-clés : Geometry of quadrangles, Urquhart's theorem 
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     author = {M. Hajja },
     title = {An {Elementary} {Proof} of "the {Most} {Elementary} {Theorem"} of {Euclidean} {Geometry}},
     journal = {Journal for geometry and graphics},
     pages = {17--22},
     year = {2004},
     volume = {8},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JGG_2004_8_1_JGG_2004_8_1_a1/}
}
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M. Hajja . An Elementary Proof of "the Most Elementary Theorem" of Euclidean Geometry. Journal for geometry and graphics, Tome 8 (2004) no. 1, pp. 17-22. http://geodesic.mathdoc.fr/item/JGG_2004_8_1_JGG_2004_8_1_a1/