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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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Another way to think about it... | Comment 11 of 21 |

Triangle ABC:  Lets put A on 0,0 and B can be on 2X,0. 

The X coord. of C needs to be on an integer and would be located at 1/2 of the distance 2X

So by definition of an equilateral triangle C will be on X,sqrt( (2X)-X )

Which simplifies to X,sqrt( 4X - X)

Which simplifies to X,sqrt( 3X).  Where sqrt( 3X) has to be an integer.

sqrt( 3X) = sqrt(3) * sqrt(X)

X has to be an integer, as it is used to define other points.  So we end up with: [sqrt(3) * integer] which isn't going to produce an integer!

And we already have the 3 space answer posted earlier...

  Posted by Gromit on 2004-06-25 02:43:22
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