Pythagorean inequality

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The Pythagorean inequality is a generalization of the Pythagorean theorem, which states that in a right triangle with sides of length $a \leq b \leq c$ we have $a^2 + b^2 = c^2$. This inequality extends this to obtuse and acute triangles. The inequality says:

For an acute triangle with sides of length $a \leq b \leq c$, $a^2+b^2>c^2$. For an obtuse triangle with sides $a \leq b \leq c$, $a^2+b^2<c^2$.

This inequality is a direct result of the Law of cosines, although it is also possible to prove without using trigonometry.

See also

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