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With a standard right triangle, we have the legs $a$ and $b$ and the hypotenuse $c$, where typically $a$ is the shorter leg. I'm curious about the fact that

$$a + b > c$$

but

$$a^2 + b^2 = c^2$$

This discussion seems to be a simple proof -- assuming the Pythagorean Theorem is true. However, at an intuitive level, it seems odd that what wasn't equal is made equal by squaring each member. Is there any theoretical explanation from higher geometry?

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  • $\begingroup$ Note $(a+b)^2\neq a^2 + b^2$ $\endgroup$ – Kulisty Oct 12 '17 at 14:58
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In $\triangle ABC$ with $\angle C$ denoting the interior angle at the vertex $ C$ and the sides' lengths being $a,b,c$ we have the Cosine Law: $$c^2=a^2+b^2-2ab\cdot \cos \angle C$$ which is very ancient. Therefore $$(a+b)^2=a^2+b^2+2ab=$$ $$=a^2+b^2-2ab \cdot \cos \angle C+2ab(1+\cos \angle C)=$$ $$=c^2+2ab (1+\cos \angle C).$$ Since $0<\angle C<\pi$ we have $1+\cos \angle C>0.$ Therefore $$(a+b)^2=c^2+2ab(1+\cos \angle C)>c^2$$ which implies $a+b>c.$

The fact that the interior angles of a triangle sum to $\pi\;$ ( implying $\angle C<\pi\;$ ) follows from the Parallel Postulate. The Cosine Law follows directly from the Theorem of Pythagoras, which follows from the Parallel Postulate, and the Theorem of Pythagoras is proved without needing to know that $a+b>c.$

The Theorem of Pythagoras can be considered a special case of the Cosine Law when $\angle C=\pi /2$ and $\cos \angle C=0.$ We can re-arrange the Cosine Law as $$\cos \angle C=\frac {a^2+b^2-c^2}{2ab}$$ from which we can see that if $\angle C<\pi /2$ then $a^2+b^2>c^2,$ and if $ \angle C>\pi /2$ then $a^2+b^2<c^2.$

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  • $\begingroup$ This is quite thorough, and I appreciate it greatly. I'm thinking that $a + b > c$ and $a^2 + b^2 = c^2$ is just "one of those things" about the math world. The former tells us the magnitudes of $a$ and $b$ are greater totaled than the magnitude of $c$, but the latter simply tells us that -- again, from a vector standpoint -- $a$ and $b$ total to $c$. So yes, two vectors total to a third vector, but their magnitudes are greater than the third vector's magnitude. Unless some "way up there" geometry can tell us how, it will simply have to be one of those curiosities of our math universe. $\endgroup$ – 147pm Oct 13 '17 at 0:23
  • $\begingroup$ There is a vast lore of equations, inequalities and other theorems about the triangle. One surprising result is Morley's theorem regarding angle trisections. $\endgroup$ – DanielWainfleet Oct 13 '17 at 5:28
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In Geometry there is an important difference between the triangle inequality $a+b>c$ and Pythagoras' theorem $a^2+b^2=c^2$: the proof of the former doesn't need the Parallel Postulate, while the proof of the latter does.

As a matter of fact, in Euclid's Elements the triangle inequality is Prop. 20 of Book I, while Pythagoras' theorem is proved much later, as Prop. 47 of Book I.

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