I am trying to find the relationship between the sides of any triangle as part of a bigger problem.
I was thinking about altering the Pythagorean theorem, but all I got was the useless $a^2+b^2\lesseqgtr c^2$ which is basically completely useless.
What I am after is something like $a^2+b^2\leq c^2$, which will actually mean something!!
The basic triangle inequality states that $$a+b\ge c$$ if $c$ is the biggest side. Triangle inequalities that are related to Pythagoras theorem are $$a^2+b^2 < c^2, \quad\text{if angle $C$ is obtuse (greater than 90°)},$$ $$a^2+b^2 > c^2, \quad\text{if angle $C$ is acute (less than 90°)}.$$ In general we may write $$a^2+b^2>\frac{c^2}{2}.$$ For more triangle inequalities, see here.