Coxeter's rabbit

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(s–b)(s–c) + s(s–c)(s–a) + s(s–a)(s–b) – (s–a)(s–b)(s–c)

Proof     s(s–b)(s–c) + s(s–c)(s–a)

          =    { algebra }

             s(s–c)(2s–a–b)

          =    { definition of s }

(1)        s(s–c)c

            s(s–a)(s–b) – (s–a)(s–b)(s–c)

          =    {algebra}

(2)        (s–a)(s–b)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.

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transcribed by Diethard Michaelis
revised Wed, 14 Nov 2007