What is the fastest proof of the Pythagorean theorem in the world?

 

Thank you

 

J5

Get out a protractor and a ruler!

That's not really proof of the theorem.

I saw a geometrical proof one time that showed two pictures of triangles bordering rectangles and that was it. Can't remember exactly what it looked like though.

 

*edit*

 

He's a pretty quick/simple one: http://www.davis-inc.com/pythagor/proof2.html

Here's a fairly quick proof.

 

Consider triangle ABC, right angled at C. Let CD be the altitude from C to AB. As usual, let BC = a, AC = b, AB = c; also let AD = x, DB = y.

 

Triangles CDB and ABC are similar; hence y/a = a/c, and a² = cy.

Triangles CAD and ABC are similar; hence x/b = b/c, and b² = cx.

 

Adding, we have a² + b² = c(x + y) = c².

Here's a fairly quick proof.

 

Consider triangle ABC' date=' right angled at C. Let CD be the altitude from C to AB. As usual, let BC = a, AC = b, AB = c; also let AD = x, DB = y.

 

Triangles CDB and ABC are similar; hence y/a = a/c, and a[sup']2[/sup] = cy.

Triangles CAD and ABC are similar; hence x/b = b/c, and b² = cx.

 

Adding, we have a² + b² = c(x + y) = c².

 

 

I went through it, wonderful proof, I won't forget it. Thanks

 

Draw the altitude, then you have created two tiny right triangles, each of which is similar to the original large right triangle. The rest follows using trig relations.

Just a thought, is it critical to use identities/equations that were not derived from the pythagoriam theorum?

Just a thought, is it critical to use identities/equations that were not derived from the pythagoriam theorum?

 

If I understand you, the answer is yes. You don't want to use the Pythagorean theorem to prove the Pythagorean theorem, that's circular reasoning which is a no no.

 

I made reference to trig relations, but they are just definitions, and they have nothing whatsoever to do with the pythagorean theorem, but definitions aren't necessary, they are for convenience.

 

Sin B = length of side opposite angle B/hypotenuse

Cos B = length of side adjacent to angle B/hypotenuse

 

So you could use them if they help you, but you don't need to have them memorized. What was needed in that proof, was the notion of similiar triangles, which doesn't have anything to do with the pythagorean theorem, and in fact proved it, as you saw. The only previous theorem that you would need, is one which taught you how to construct the altitude, and that is covered in Book I of Euclid, long before Euclid proves the Pythagorean theorem.

 

Here it is:

 

Euclid's twelfth proposition

Seriously.

Proving the pythagorean theorem?

all you really need is a pencil, paper, and a ruler. You don't even have to have knowledge of trigonometry.

 

I uploaded an attachment. IN the attachment is an incredibly easy to follow proof of the pythagorean theorem. Any questions, please respond.

Pythagorean Theorem.doc