How do squares of non-right triangles relate?

Question

How do squares of the sides of a triangle, any triangle, relate?

Answer 1

The general rule that relates the sides of a triangle is called the law of cosines. It regards the following situation, where you know an angle of the triangle and want to relate the side lengths:

It is stated as follows: $$C^2=A^2+B^2-2AB\cos(\alpha).$$ Note that this has a very similar form to the Pythagorean theorem - just with an extra adjustment for the angle $\alpha$. In particular, if $\alpha$ is a right angle, this last term is zero, so it reduces to the Pythagorean theorem. The equation also tells us that when $\alpha$ is obtuse, the opposite is longer than it would be in a right triangle and when $\alpha$ is acute, the opposite side is shorter.

Answer 2

Sorry for the short answer.

Cosine Rule, whose special case is the Pythagoras theorem.

$$ c^2= a^2 + b^2 - 2 a b \cos C$$

when $C = 90^0. $