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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 :)

curious
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    It will be more fun if we have that topic delayed till the introduction of the cosine law. – Mick Apr 28 '16 at 13:44

1 Answers1

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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.

Siddharth Bhat
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  • Yes, of course with the use of trigonometry one can say even more, but is trigonometry part of elementary geometry as Pythagoras' is? – curious Apr 28 '16 at 13:48
  • Yes. It may depend on the country, but when I was in middle school it was introduced the same year as Pythagora's theorem. – Captain Lama Apr 28 '16 at 13:53