Pythagoras

We can get so used to the Pythagoras theorem, after all they make us use it again an again in school, but now lets stop and think for a moment. Why the hell it is true??!!

To see what I am mean try to say with a straight face that:

"It absolutely clear and intuitive that:

Hypotenuse ² = (One leg) ² + (Other leg) ² "

Well if you can't say that you have company! Me at least. And if you can please try to convince me!!

There are many different demonstrations for Pythagoras, in this post I will point to some of the more famous ones, some less known, and different/unusual ones.


One thing that bothers me is that squared in the formula, what do squares have to with it any way?

Notice that 3 ² = area of the square of side 3=9, that is why we say "raised to the power two" as "squared"

So we can rephrase Pythagoras as:

The sum of the areas of the two squares over the legs equals the area of the square over the Hypotenuse.

The first demonstrations actually demonstrated that area version of Pythagoras.

Lets begin with the famous ones. Here I will shamelessly point to the collection at Cut the Knot, I will comment some of these.

#4 is an area argument, pretty straightforward. Lots of people like it but doesn't appeal a lot to me.

#6 is the first I ever saw, it uses the similarities with the small triangles formed by the height, you get the same intermediate steps of Euclid's # 1 on the list.

A more honest proof since it uses something we are more used too but kind of tedius to find the right relations to use.

first show a ² m*c
and b ² n*c

and add them together to get a ² + b ² = c *(m+n)=c*c=c ².

#7 Is the second demonstration of Euclid, takes some thinking to understand. But as my friend said you can't hope to get better than that(in some sense).

#9 Another area argument pretty nice (no formulas!!)

#12 is a more 'mechanical' proof (one of my favorites)

#15 is probably the best area argument (uses two different tilings of the plane the two square and the 'diagonal' square) to prove that they have the same area, go here for anothere explanation.

#40 is a "infinitesimals proof" pretty nice if you know calculus.

#41 That is really cheating!!

#43 Is a circle demonstration (prefer that than #11).

Observation 11 before the proofs is a nice generalization of Pythagoras by Edsger W. Dijkstra. Using an idea somewhat similar to demon. #6 with a nice twist.


You can also look at the others demonstrations at Cut the Knot and make your own judgments.

In the next posts I will try to give some different demonstrations, here is a post with a "physical" demonstration.