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?
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)²-X² )
Which simplifies to X,sqrt( 4X² - X²)
Which simplifies to X,sqrt( 3X²). Where sqrt( 3X²) has to be an integer.
sqrt( 3X²) = sqrt(3) * sqrt(X²)
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...
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Posted by Gromit
on 2004-06-25 02:43:22 |
Another way to think about it...
