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Is it possible to construct an equilateral triangle with coordinates on a grid of integers?

I think the answer is no, but how can I prove this?

I started with a triangle with coordinates (0,0) (a,b) and (c,d). Equating the size of the 3 sides, I get

$a^{2}+b^{2}=c^{2}+d^{2}=2ab+2cd$

How should I continue?


I see there are solutions based on the fact that the angle between two edges can not be 60°. Is it possible to have a solution based on the fact that the length of the edges can not be the same?

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This deals with a general version of your question. In particular, since $\tan\frac{\pi}{3}$ isn't rational, you can't have lattice points as the corners of an equilateral triangle. –  J. M. Feb 3 '12 at 15:02
    
possible duplicate of Which internal angles can a lattice polygon have? –  joriki Feb 3 '12 at 15:09
    
You've accepted an answer which I believe is incorrect. Please see my comment under the answer, and unaccept it in case you agree, as the checkmark will otherwise mislead others. –  joriki Feb 3 '12 at 15:23
    
$(a-c)^{2}+(b-d)^{2}=a^{2}+b^{2}+c^{2}+d^{2}-2ab-2cd=2a^{2}+2b^{2}-2ab-2cd$ this implies $a^{2}+b^{2}=2ab+2cd$ –  wnvl Feb 3 '12 at 15:25
    
a) If you edit your post such that a correct earlier comment by someone else now appears wrong, please indicate this clearly (e.g. by adding "Edit" or the like). b) It's still wrong, since you've shown that $a^{2}+b^{2}=2ab+2cd$, whereas the post now says $a^{2}+b^{2}=2ac+2bd$. –  joriki Feb 3 '12 at 15:30

4 Answers 4

up vote 10 down vote accepted

Let the vertices of our triangle be $(0,0)$, $(a,b)$, and $(c,d)$, where $a$, $b$, $c$, and $d$ are integers. If all edge lengths are the same, then $$a^2+b^2=c^2+d^2=(a-c)^2+(b-d)^2.$$ Minor manipulation turns this into $$a^2+b^2=c^2+d^2=2ac+2bd.$$

Now we use my favourite identity, which was known more than a millenium ago in India, and even earlier by Diophantus, and so has often been called the Fermat Identity: $$(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2.\qquad\qquad(\ast)$$ This identity can be easily verified by expanding both sides, or more conceptually by noting that the norm of the product of two complex numbers is the product of the norms.

Let $N=a^2+b^2=c^2+d^2=**2(ac+bd)**$. Then $ac+bd=N/2$. The identity $(\ast)$ now gives $$N^2=\frac{N^2}{4}+(ad-bc)^2$$ or equivalently $$3N^2=4(ad-bc)^2.$$ This is impossible, since $3$ times the perfect square $N^2$ cannot be a square unless $N=0$, which gives a very tiny triangle.

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A cute proof of the fermat identity: If $z = a+ib$ and $w = c + id$, then $ (zz') (ww') = (zw)(zw)'$ where $x'$ is the conjugate of $x$. –  Aryabhata Feb 3 '12 at 21:05

It is possible, but you need three dimensions in order to do it.

Consider $\bigtriangleup v_{1}v_{2}v_{3}$ with:

$v_{1}=(1,0,0)$

$v_{2}=(0,1,0)$

$v_{3}=(0,0,1)$

For $a,b\in{1,2,3}$, $a\neq b$, $d(v_{a},v_{b})=\sqrt{2}$, therefore the triangle is equilateral. It is not possible (as other answers indicate) to have an equilateral triangle with integer coordinates for the vertices in a two dimensional square lattice (a grid is just a 2d lattice).

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Solution 1 (by me): Assume WLOG that two of the points are $(0,0), (m,n), m,n \in \mathbb{Q}$. Then the third point is $(m/2 - n \sqrt{3} / 2, n/2 + m \sqrt{3}/2)$, which is not a rational point.

Solution 2 (by a friend): The determinant formula for area is rational, so if the all three points are rational points, then the area of the triangle is also rational, so whereas the area of an equilateral triangle with side length s is $\frac{s^2 \sqrt{3}}{4}$, which is irrational since $s^2$ is an integer.

Note that the above solutions both generalize from integer points to rational points.

You can also use Pick's theorem for integer points.

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You can start like this. Without loss of generality, let the three points be $(0,0), (0,a)$ and $(b,a/2)$. You can do this because you can always rotate and translate the axis to get these points. Now for all the points to be integral, you need $a$ to be even. This is the first constraint. Secondly, from the basic trigonometry,

$$ \tan \theta = \frac{2b}{a} = \sqrt{3}. $$ From this, you get $b = a\sqrt{3}/2$, which is irrational.

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You can't isometrically map any three lattice points to points of this form. The lengths of all sides may be irrational, whereas the length of the side from $(0,0)$ to $(0,a)$ is always an integer. –  joriki Feb 3 '12 at 15:17
    
Ohh yes! Thanks for pointing it out. –  Jalaj Feb 3 '12 at 22:43

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