__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*)(2*s*–*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.

(End of Proof)

Nuenen, 14 April 2002

prof.dr.Edsger W.Dijkstra

Plataanstraat 5

5671 AL Nuenen

The Netherlands

transcribed by Diethard Michaelis

revised