Pythagorean theorem

Geometry

Statement

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

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

A right triangle with sides a, b and c

Proof

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

A square with four identical triangles inside it

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)