The axiomatic structure of geometry, as initiated by Euclid and then developed by other mathematicians starts of with 8 axioms or postulates which are self-evident truths". Chains of logical reasoning can be used to prove theorems which are then accepted as additional truths, and so on. Geometry does not have laws, as such.
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Yes. His name is on it.
Hmmm... because HE created it ! Pythagorean theorem is named after the mathematician Pythagoras ! See related link for the Wikipedia article on the gentleman concerned !
If you have two lengths of the triangle, the Pythagorean theorem will help you find the third. As it is, you need to find two numbers whose squares add up to 1296. There are a lot of possibilities.
If, by trigonometry theorem you mean the "fundamental theorem of trigonometry," sin2(x) + cos2(x) = 1, it is actually the Pythagorean Theorem. if you have a right triangle with a hypotenuse of one, sin(x) is one leg, and cos(x) is the other. The Pythagorean Theorem states that a2 + b2 = c2 and therefore sin2(x) + cos2(x) = 1.
There are a great number of different proofs of the Pythagorean Theorem. Unfortunately, many of them require diagrams which are hard to reproduce here. Check out the link to Wikipedia's page on the theorem for several different proofs.