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Coxeter's rabbit

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On p.13 of his Introduction to Geometry, H.S.M. Coxeter invites the reader to see (and to use spontaneously) that with s = (a+b+c)/2 , abc equals

(0)   s(sb)(sc) + s(sc)(sa) + s(sa)(sb) – (sa)(sb)(sc)

Proof     s(sb)(sc) + s(sc)(sa)

          =    { algebra }

             s(sc)(2sab)

          =    { definition of s }

(1)        s(sc)c

            s(sa)(sb) – (sa)(sb)(sc)

          =    {algebra}

(2)        (sa)(sb)c

Because both expressions (1) and (2) contain a factor c , so does (0); for reasons of symmetry, (0) also contains factors a and b , i.e. is a multiple of abc. The coëfficient equals 1 —as is trivially established with, say, a,b,c := 2,2,2 — and thus abc = (0) has been proved.

(End of Proof)

Nuenen, 14 April 2002

prof.dr.Edsger W.Dijkstra
Plataanstraat 5
5671 AL Nuenen
The Netherlands


transcribed by Diethard Michaelis
revised Wed, 14 Nov 2007