Given a triangle , we can write any planar point where

Furthermore:

Where are the ratios of the signed areas

Proof

We'll prove this for a triangle, but a similar proof generalizes for a simplex of any dimension

We can subtract from both sides to get which treats as the origin and highlights that we only have two degrees of freedom. So to simplify:

Which gives and can be written in matrix form

This can be solved using Cramer's rule:

But recall that each determinant is the signed area of the parallelogram spanned by its column vectors and so

To show that consider that given any non-degenerate triangle and a fixed planar point we can compute the signed area: