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:
By Heron’s formula, the area of triangle in Fig. 1
where are three sides of the triangle and, .
Let , we have
Compute using Omega CAS Explorer (see Fig. 2) , the result shows
Hence, (1) and (2) are equivalent.
I would like to learn any other alternative proof.