Cool graphic proofs

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.

Triangular numbers

In number theory, the sum of the first ncubes is the square of the nth triangular number. That is,

1^3 + 2^3 + 3^3 + \cdots + n^3 = \left( 1 + 2 + 3 + \cdots + n \right)^2.

The same equation may be written more compactly using the mathematical notation for summation:

\sum_{k=1}^{n} k^{3}= \left(\sum_{k=1}^{n}k \right)^{2}.

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.

Binomial theorem


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.


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