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Lattice Points and Equilateral Triangles

Date: 12/08/2002 at 12:08:34
From: Joe
Subject: Lattice points and Equilateral triangles

My Geometry teacher assigned a project for equilateral triangles. He 
wanted our answers to be proofs. He wanted as to prove whether or not 
an equilateral triangle could have lattice points as its vertices.

I understand that the answer is no; however, he mentioned that there 
were three scenarios for the positions of the triangle. They are: 
base is vertical, base is horizontal, and base is oblique. He is 
expecting three different formulas (one for each scenario). I cannot 
find the formulas. Also, would my answer really change for each 
scenario? 

I have already figured out the following:

   - the equilateral triangles are made up of two congruent right 
      triangles
   - the only time a right triangle's hypothenuse is even is when 
      it is a 3,4,5, etc.
   - when it is a 3,4,5 type the two cannot make an equilateral
      triangle

Please help me.
Thank you for your time.
Joe


Date: 12/08/2002 at 14:42:23
From: Doctor Floor
Subject: Re: Lattice points and Equilateral triangles

Hi,

Thanks for your question.

I will try to explain how you can prove the third case (base is 
oblique). We will use that sqrt(3) is irrational; see for instance 
from the Dr. Math library:

   Proof that Sqrt(3) is Irrational
   http://mathforum.org/library/drmath/view/52631.html 

So we will try to construct an equilateral triangle ABC of lattice 
points.

We may assume that A(0,0) is one of the lattice points, and B(2t,2u) 
is one of the others. Then the midpoint of AB is M(t,u). Now from M 
to the third vertex C we have to go along a line perpendicular to AB 
through a distance of sqrt(3)*sqrt(t^2+u^2).

Since the slope of AB is u/t, and the product of slopes of 
perpendicular lines must be -1, we know that the slope of MC must be 
-t/u. Or stated in another way: we have to add to the coordinates of 
M a multiple of (-u,t).

Since the distance from M to C has to be sqrt(3)*sqrt(t^2+u^2), we 
see that in fact we have to add +/-sqrt(3)*(-u,t). And now we see 
that the coordinates of C cannot be rational, and thus C cannot be a 
lattice point.

If you look very carefully, you will see that this method applies to 
cases 1 and 2 as well.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/ 


Date: 12/08/2002 at 15:12:22
From: Joe
Subject: Thank you (Lattice points and Equilateral triangles)

Thanks so much! I actually understand it now. You guys rock!
Associated Topics:
High School Coordinate Plane Geometry

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