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Equilateral Triangle (Posted on 2004-06-20) Difficulty: 4 of 5
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.)
re(3): short answers to a short q. | Comment 9 of 21 |
(In reply to re(2): short answers to a short q. by Ady TZIDON)

(0;0;0)(2a;0;0)(a;a;2*a) does not have sides of equal length.

Also, proving that it can't be done in 2D cannot be dismissed so lightly.  That the square root of 3 is irrational certainly does not by itself prove the result; a lot more work is needed like what Tristan gave below.  In fact, the sides of the 3D equilateral triangle with vertices (1,0,0), (0,1,0), (0,0,1) all have the irrational length of square root 2.

Edited on June 22, 2004, 1:37 pm
  Posted by Richard on 2004-06-22 13:04:33

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