Has anyone a funny / interesting / pedagogical proof or illustration

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only one I ever understood

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i would tell you but i'd have to throw you off this boat afterwards

>>8383830

Not a proof but i want to be math

3^2 + 4^2 = 5^2

#euler

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>>8383847
very nice, thank you. very simple to understand at all ages.
I found this one. you have to be more knowledgable (similar triangles, algebraic identities)

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Given:
a, b in R^n
ab = 0

Then:
|a+b|^2 = (a+b)(a+b) = aa + 2ab + bb = |a|^2 + |b|^2

QED

>>8383850
i like the fact that he is holding a pyramid and not a triangle

>>8383847
There's a gif with water-filled squares that do the job too.

>>8383894
THAT'S NOT A PROOF YOU IDIOT

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This isn't cheating

>>8383897
of course it is
YOU are the idiot

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My favorite one (thanks Gilbert S.)

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>>8383931
could you explain area=(bc)ac/2 ?

>>8383958
That final triangle is a right triangle, so the base*height/2 could also be written as bc*ac/2. The picture didn't really show how it's a right triangle, but since the two smaller triangles are similar, and you connect them via complementary angles, the two angles add to 90.

>>8383977
yep ok thanks

>>8383847
/thread

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>>8383830
Proofs without words are objectively the best proofs.

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>>8384187
this one is ancient, in a particular case.
you're right about proof without words. classy and clear. no beat.
here's euclide's scheme for his proof. used to be taught in school. isn't anymore.

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Take any right triangle, and call the length its one leg [math]a[/math] and the other [math]b[/math], and the hypotenuse [math]c[/math]. Construct a square with side [math]a+b[/math] by placing the triangles so that every angle opposite to [math]a[/math] meets precisely one angle opposite to [math]b[/math]. Now, the area of the square is [math](a+b)^2[/math], but also [math]4(\frac{1}{2}ab) + c^2=2ab+c^2[/math]. Thus, [math]a^2+2ab+b^2=2ab+c^2 \Leftrightarrow a^2+b^2=c^2[/math].

This is probably the most clear proof for the Pythagorean theorem I've seen. It's not funny or anything, but is a purely algebral solution to a geometrical proof. But, it has been said that geometry is drawn algebra and algebra is written geometry. This proof is an example of that.

>>8383889
...

We are in a Native American camp. There are three hides on the ground. On a hippopotamus hide sits a squaw who weighs 200 pounds. On a buffalo hide sits a squaw whose son weighs 100 pounds. On a bear hide sits a squaw whose son also weighs 100 pounds.

The squaw on the hippopotamus is equal to the sons of the squaws on the other two hides.

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>>8383830

Of course, as the OP has tacitly recognized by his question, the Pythagorean theorem has many distinct proofs, dozens, if not hundreds.

A historically interesting proof of PT, as I shall call it, was made in 1876 by United States president James Garfield while he was still in the House of Representatives. Per wiki, this proof was published in The New England Journal of Education. This makes Garfield a US president who was also a published mathematician.

Unfortunately for Garfield, he was assassinated in 1881 by a looney named Charles Guiteau just a few months after he took office, and then Garfield lingered very painfully for several weeks before finally succumbing. Alexander Graham Bell attempted to locate the bullet with a crude metal detector, bu the schmucks failed to account for the fact that Garfield's bed had (wait for it...) a metal frame. Garfield's last few years therefore constitute an interesting curiosity in the history of science.

But on to Garfield's proof.

http://io9.gizmodo.com/james-garfield-was-the-only-u-s-president-to-prove-a-m-1037750658

The basic idea of the proof is roughly "half" of what is done in >>8383847 .

1) Take any right triangle that you wish, and make a copy of it. Lay those two copies opposite-corner-to-opposite corner as in pic related, so that one triangle's leg a is colinear with its copy's leg b.

2) Make the requisite observations to establish that that "interior" angle of the "c-c" triangle is always a right angle.

3) Recall the high-school tier formulae for the areas of triangles and trapezoids, which apply in this case. Rearrangement forces us to conclude le ebin Pythagorean Theorem! xD

This would be a great thing to teach in a high-school setting, because it pulls everything together: a little bit of history, thinking about how proofs work and why, and recycling the area formulas with which students are already familar with in service of something more complex/interesting, which is the proof itself.

>>8385183
ahahah

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>>8383830
mhmm i like this method

express the square as both a total object and as a part of a smaller square with 4 triangles added...

>>8385736

1) (a+b)^2= a^2 +2ab+b^2

2) c^2 +4(1/2)ab = c^2 +2ab

set 1 and 2 equal to eachother since we are talking about the same total area

c^2 +2ab = a^2 + 2ab + b^2

c^2 = a^2+ b^3

>>8385737
woops
\
c^2 = a^2+ b^2

There you are (from Numberphile)
https://youtu.be/ItiFO5y36kw?t=7m30s