# Pythagorean theorem

## Statement

In a right triangle with the hypotenuse side being called $c$ and the other sides $a$ and $b$, the following equation applies:

$$a^2 + b^2 = c^2$$

## Proof

Draw a square and four identical triangles inside it. Now the larger square contains a smaller square, like this:

The area of the large square is $(a + b) * (a + b) = (a + b)^2$.

The area of the smaller square is $c * c = c^2$.

The area of one of the triangles is $\frac{1}{2} * a * b$. Thus the area of all four triangles combined is $4 * (\frac{1}{2} * a * b) = 2ab$.

Note that the area of the bigger square is the same as the area of the four triangles and the area of the small square.

$$\overbrace{(a + b)^2}^\text{large square} = \overbrace{2ab}^\text{four triangles} + \overbrace{c^2}^\text{small square}$$

When you expand the left hand side, you get the following.

$$a^2 + 2ab + b^2 = 2ab + c^2$$

Subtract $2ab$ from both sides to get the final result.

$$a^2 + b^2 = c^2$$

And hereby the Pythagorean theorem is proven.

## Proofs building upon this proof

### Cosine rule

This proofs shows that the cosine rule is correct.

### Distance formula between a point and a line

This proof shows a formula for calculating the distance between a point and a line in the form of y = ax + b.

### Heron's formula: area of triangle

This shows Heron's formula for finding the area of any triangle.

### Sine squared plus cosine squared is always one

This proof shows that sine squared plus cosine squared is always one, using the Pythagorean theorem.

### Trig values of 18 degrees

This proof shows the exact value of sin(18), cos(18) and tan(18)

### Trig values of 30 and 60 degrees

This proof shows the exact value of sin(30), cos(30), tan(30), sin(60), cos(60) and tan(60).

### Trig values of 45 degrees

This proof shows the exact value of sin(45), cos(45) and tan(45)