TITLE: generalization of the Pythagorean theorem
QUESTION [1 upvotes]: In school, students learn that in a triangle ABC, ACB is a right angle if and only if AB^2=AC^2+BC^2. This deep relation between geometry and numbers is actually only a partial result as one can say much better : the angle in C is

acute if and only if AB^2 < AC^2+BC^2, 
right if and only if AB^2 = AC^2+BC^2,
obtuse if and only if AB^2 > AC^2+BC^2.

I was wondering why this relation is unknown to almost all people and never taught in school. Does it require more mathematic understanding? does it require analytic geometry? Thanks in advance for comments :)

REPLY [0 votes]: The relation can actually be thought of in terms of the cosine rule:
$a^2 = b^2 + c^2 - 2bc \cos(A)$, where $a, b, c$ are the sides of the triangle and $A$ is the angle opposite to side $a$.
Clearly, if $A = 90^\circ$, then $a^2 = b^2 + c^2$
If $A < 90^\circ$, then $\cos(A) > 0$, hence $a^2 < b^2 + c^2$ and vice versa if the triangle is obtuse.
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