# The most beautiful formulae/theorems/identities in mathematics

This is my personal collection of formulae/theorems which I consider lovely. By “lovely”, I mean objects which possess a certain degree of Elegance and Simplicity. The formulae/theorems are listed in no particular order.

**Pythagoras’ theorem**

The most popular and fascinating theorem in Euclidean geometry takes the first place in the list.

If AB, BC and AC are three sides of a right angled triangle ABC, where AC is the hypotenuse, then

**Euler’s formula**

, where is the Euler’s number.

**Heron’s formula**

. where is the area of a triangle whose sides are of length and perimeter is .

**Bayes theorem**

Or

**Sine rule**

If A,B,C are vertices of a triangle, and sides a,b,c are a = BC, b = CA, c = AB then

**Cayley – Hamilton theorem**

every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

**Euclid’s algorithm**

If a and b are integers and a > b, then gcd(a, b) = gcd(a (mod b), b)

**Trigonometric gem 1**

sin(x – y) sin(x + y) = (sin(x) – sin(y)) (sin(x) + sin(y))

**Trigonometric gem 2**

X+Y+Z = X*Y*Z if

X = tan(A)

Y = tan(B)

Z = tan(C)

and

Of course, this list is undeniably incomplete. There will be more entries, as I discover more gems.