This post is supposed to collect cool geometric proofs that I’ve found on other websites. The current version contains a proof of Nicomachus’s theorem and of the Binomial theorem.
In number theory, the sum of the first ncubes is the square of the nth triangular number. That is,
The same equation may be written more compactly using the mathematical notation for summation:
This identity is sometimes called Nicomachus’s theorem.
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.
For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a×a×b rectangular boxes, and three a×b×b rectangular boxes.