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 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

Invalid username
Login to AoPS