Heron’s formula concerning , the area of any triangle states:

where a,b,c are the three sides of the triangle and, .

We are going to prove it with the aid of a CAS:

Substituting into (1), the formula becomes

A triangle with three known sides is shown in Fig.1 where is part of the base of the triangle.

Fig. 1

By Pythagorean theorem,

To obtain , we will use Omega CAS Explorer (see Fig. 2)

The function ‘eliminate’ eliminates variable , returns the value of

Fig. 2

The result is , i.e.,

.

Therefore, using the standard formula for triangle,

which is (2)

This is the 1st example in my presentation at ACA 2013 titled “An Algebraic Approach to Geometric Proof Using a Computer Algebra System”.

In Fig. 3, . Can you find the area of ?

Fig. 3