Pythagorean Inequality (Hinge Theorem Derivation)

Hopefully you are familiar with the pythagorean which is:

$a^2+b^2=d^2$

Where a, b, and d are sides of a right triangle and d is the hypotenuse.

Now what happens if we decide to change this right angle? Let's find out. So if we decrease this right angle we result in a acute triangle. So the length of sides a and b are the same but side d has new length. Let's call this c.

Quite obviously $c<d$, this results in $c^2<d^2$ so ultimately we have the first part of the pythagorean inequality which is: if $c^2<a^2+b^2$ then triangle abc is acute.

What if we increase this angle. By similar methods we arrive at $a^2+b^2<c^2$ for obtuse triangle abc.

So to recap the pythagorean inequality is:

For right triangle: $a^2+b^2=c^2$

For acute triangles: $c^2<a^2+b^2$

For obtuse triangles: $a^2+b^2<c^2$