comparisons of side lengths on a triangle

Question

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!!

um @vitamind i kind of need an equality that tells me something about the sides, whilst $\lesseqgtr $ just tells me that $a^2+b^2$ and $c^2$ can be anywhere. — Arale, Feb 22 '21 at 12:52
There is the inequality: $x+y\ge z$, don't know if it helps you. — vitamin d, Feb 22 '21 at 12:54
Yes but $z$ has to be the longest side. https://en.wikipedia.org/wiki/Triangle_inequality — vitamin d, Feb 22 '21 at 12:58

score 1 · Accepted Answer · answered Feb 22 '21 at 13:08

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.

comparisons of side lengths on a triangle

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