Pythagorean inequality

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 states:

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