Proof of Heron's Theorem

Consider a triangle ABC as shown below:

First the altitude h from the vertex opposite c is constructed.


Now side c is divided into two sections, one of length x and the other of length c - x as shown.


We can use the Pythagorean theorem to find h in terms of a, b, x, and y.

For the different parts of the triangle, we have that:


(2) and

(3) by construction

Considering that the area = , then the area squared = , so .

Rearranging equation (3) and substituting into equation (2) yields the following result:

Subtracting this result from equation (1) yields:

Rearranging equation (1) and multiplying through by yields the following:

which factors into

where gives the following:

which can be rearranged to yield:

Since , this equation becomes:

Factoring out the two's and dividing on both sides results in this equation:

Taking the square root of both sides results in Heron's formula for the area of a triangle: