If a triangle has side lenghts $a,b,c$ where $c$ is the largest prove that its obtuse if $c^2>a^2+b^2$ and acute if $c^2<a^2+b^2$.

Question

I was thinking about this and I cant get to a formal proof. I have a sort of mental image where you draw $a$ and $b$ perpendicular and the $c$ is too small to connect the two endpoints. So the right angle enclosed by $a$ and $b$ has to be reduced a bit so that $c$ can connect the end of $a$ to $b$. But this is not really a proper proof. I searched the internet and this idea is used in a lot of places but nowhere proven. This is not homework, but I was writing a little program to determine whether a given triangle will be obtuse, acute or right-angled. Any help would be much appreciated :) Thanks in advance.

Answer 1

The "law of cosines" tells us that $$a^2=b^2+c^2-2bc \cos A$$This is a useful generalisation of Pythagoras to triangles which are not right-angled, and validates your observation.

Note that noticing things which are not immediately obvious is one of the most powerful ways in which mathematicians make progress!!