Pythagorean theorem proof and Euclid

Today a group did a presentation on the Pythagoras and the Pythagorean theorem proof. Their proof for the theorem is as followed:
Proof:
We have a right triangle

and with the hypotenuse we can construct a square with side length c. The the Area( big square) = c^2. Within the square, the right hand triangle can be placed in such a way that it can fit into the square 4 times.

The area of the big square can be also be calculated by finding the area of the individual shapes and summing them together.
Area(big square) = 4*Area(triangle)*Area(small square)
                   C^2 = 4[(1/2)ba] *[(b-a)(b-a)]
                          = 2ba*[(b^2)-2ba+(a^2)]
                         = a^2 +b^2

Thus c^2 = a^2 +b^2                                                                                           QED

Little is know about Euclid of Alexandria, a Greek mathematician; he is most famous for his book, Elements, which is broken down into 13 books. Elements is responsible for motivating famous mathematicians to become mathematicians. In book I, Euclid proves various theorems, all which help him prove a proposition at the end (the Pythagorean theorem). His proof of the Pythagorean theorem is different from the one mention above.

Euclid's proof of Pythagorean theorem

Some proposition Euclid also proved include

1. One a given straight line to describe a square
2. If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal sides also equal, then the two triangles are congruent.
3. If a straight line falling on two straight lines makes the alternate angles equal to one another, then the straight lines are parallel to one another.

 In one of the proposition, prop I- 41, Euclid uses the phrase "in the same parallels" and in modern terms it would mean that two figures have the same height. The author, Katz, stated that Euclid, on occasion, relied on diagrams more than modern mathematicians would allow.

Unanswered question:

1. What other proofs of the Pythagorean theorem exist?

Evaluation:
Interesting: 4
quality:7
complexity: 6