Lattice Points and Equilateral TrianglesDate: 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! |
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