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This is basically related to the required angle of 60°. The tangent of 60° is an irrational number; thus, if one of the sides is horizontal, and one of the triangle's endpoints is in the origin of coordinates, you won't find integer numbers that lie directly on the 60° line. If you start from a non-horizontal position, you would still run into the same problem, since the "base side", the first side you draw, has integer-valued endpoints - and the tangent of the angle will be rational. You'll find a more detailed proof in several places online, but that's the basic idea.

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