The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs. In other words, if c is the length of the hypotenuse, and the two legs have length a and b, then
c² = a² + b²
A visual proof of this theorem follows:
Here you see three squares, with areas equal to the squares of the three sides of the red triangle. The two smaller squares can be rearranged to make the larger square, proving the sum of the two smaller squares is the larger square.
Integer values of a, b, and c that make this equation true are called Pythagorean Triples.
Go back to the Number Theory Home
Formulas for Primitive Pythagorean Triples and their deriviation -- a way to generate all the triples such that a^2 + b^2 = c^2
Prove that the area of a right triangle with integer sides is not a perfect square. The proof is here (with some help from someone from the nrich website)
Puzzle question -- if m is the product of n distinct primes, how many different right triangles with integer sides have a leg of length m?
Arithmetic Sequence of Perfect Squares, page 3 -- If a^2, b^2, c^2 are in arithmetic sequence, why is their constant difference a multiple of 24? Look at the second answer to this question for the relationship between Pythagorean Triples and this arithmetic sequence of squares.
Theorems Involving Perfect Squares -- answers questions such as "why is the square root of x irrational unless x is a perfect square?" and other fundamental questions about perfect squares.
The webmaster and author of this Math Help site is Graeme McRae.