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This triangle is ~similar(geometrically) to the 1:2:sqrt(3) right triangle. Since sqrt(3) is irrational you can’t have all three sides be positive integers. You can prove sqrt(3) is irrational by negation: assume sqrt(3) = p/q for p and q coprime and nonzero, then p^2=3q^2 so p is divisible by 3 so p=3k for some k so 9k^2=3q^2 so 3k^2=q^2 so q is divisible by 3 so p and q aren’t coprime which negates the assumption so sqrt(3) does not = p/q.
