perplexus.info :: Geometry : Equilateral Triangle

Equilateral Triangle (Posted on 2004-06-20)

Can an equilateral triangle have vertices at integral lattice points?

Integral lattice points are such points as (101, 254) or (3453, 12), but not points such as (123.4, 1) or (√2, 5)

If you can't find a solution in the 2D Cartesian plane, can you find one in a 3 (or more) dimensional space?

No Solution Yet Submitted by SilverKnight

Rating: 2.6000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

2D No - 3D ?

| Comment 14 of 22 |

For 2D the answer is no. Assume true with s the length of an edge. The area of the triangle is s^2*sqrt(3)/4. By Pick's theorem the area is k/2 for some integer k. Therefore,

sqrt(3) = 2*k/s^2

s^2 is an integer for vertices lying on latice points. Hence, we have a contradiction since sqrt(3) is not rational.

Posted by Jerry on 2004-07-17 13:49:49

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