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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)

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Solution Proof | Comment 5 of 21 |

Let's assume that there is such an equilateral triangle with integral lattice points.  Call it triangle ABC, and set point A as the origin.

Extend BC to double its length to point P.  APB is a 30-60-90 triangle with integral lattice points.  Note that AP=sqrt(3)*AB.

| \
| \
| C
| / \
| / \
|/ \


Let's call the coordinates of B (X,Y).  P's coordinates must be (ħsqrt(3)*-Y,ħsqrt(3)*X) because the direction (vector?) from A is rotated 90 degrees and the magnitude is sqrt(3) times more.  Since X and Y are integers, P cannot be an integral lattice point.  This leads to a contradiction, so therefore, there is no such triangle.

As for part 2, (1,0,0), (0,1,0), (0,0,1) is the smallest such equilateral triangle, but there are many more oriented in different ways.

  Posted by Tristan on 2004-06-20 15:52:09
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