In my previous two posts, “An Algebraic Proof of Heron’s Formula” and “An Alternative Derivation of Heron’s Formula,” I proved Heron’s formula for the area of a triangle with three given sides. Based on Heron’s formula, we can now prove a theorem concerning the area of any triangle in a rectangle coordinate system, namely,

The area of a triangle with vertices at in a rectangle coordinate system can be expressed as

where is the determinant of matrix:

I offer the following proof:

Fig. 1

By Heron’s formula, the area of triangle in Fig. 1

where are three sides of the triangle and, .

Therefore,

where

Let , we have

Compute using Omega CAS Explorer (see Fig. 2) , the result shows

Fig. 2

Since implies

implies

,

i.e.,

Hence, (1) and (2) are equivalent.

I would like to learn any other alternative proof.