From Courant's Differential and Integral Calculus p.13,(adsbygoogle = window.adsbygoogle || []).push({});

In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are calledlattice points.Prove that a triangle whose vertices are lattice points cannot be equilateral.

Proof: Let ##A=(0,0), B=(\frac{a}{2},b), C=(a,0)## be points in a rectangular coordinate system where ##a## and ##b## are integers. Equilateral triangles have angles 60° at every corner, and we know that ##tan(60)=\sqrt3##. So, the distance from the point ##B## to the line ##\bar {AC}## is ##\frac{a}{2}tan(60) = \frac{\sqrt3}{2}a## which is an irrational number and contradicts the fact that the distance from the point ##B## to the line ##\bar {AC}## is an integer.

But according to other post and resources this proof is flawed. Can anyone help me with this?

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# I Proof that lattice points can't form an equilateral triangle

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