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190 geometry/lattice/equilateral.p




Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

190 geometry/lattice/equilateral.p


Can an equlateral triangle have vertices at integer lattice points?

geometry/lattice/equilateral.s

No.

Suppose 2 of the vertices are (a,b) and (c,d), where a,b,c,d are integers.
Then the 3rd vertex lies on the line defined by

(x,y) = 1/2 (a+c,b+d) + t ((d-b)/(c-a),-1) (t any real number)

and since the triangle is equilateral, we must have

||t ((d-b)/(c-a),-1)|| = sqrt(3)/2 ||(c,d)-(a,b)||

which yields t = +/- sqrt(3)/2 (c-a). Thus the 3rd vertex is

1/2 (a+c,b+d) +/- sqrt(3)/2 (d-b,a-c)

which must be irrational in at least one coordinate.

 

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