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,

- .

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

- .
This identity is sometimes called

Nicomachus’s theorem.

https://en.wikipedia.org/wiki/Squared_triangular_number

A

triangular numberortriangle numbercounts objects arranged in an equilateral triangle, as in the diagram on the right. Thenth triangular number is the number of dots composing a triangle withndots on a side, and is equal to the sum of thennatural numbers from 1 ton.

# Binomial theorem

For positive values of

aandb, the binomial theorem withn= 2 is the geometrically evident fact that a square of sidea+bcan be cut into a square of sidea, a square of sideb, and two rectangles with sidesaandb. Withn= 3, the theorem states that a cube of sidea+bcan be cut into a cube of sidea, a cube of sideb, threea×a×brectangular boxes, and threea×b×brectangular boxes.

https://en.wikipedia.org/wiki/Binomial_theorem#Geometric_explanation