Caveat: I know little about geometry but as I learn I uncover unnoded material. I don't see why barycentric coordinates are important but I will present their definition and prove the coordinates are unique.
Theorem: Let A, B, and C be vectors defining three non-collinear points. Then any point D in the plane defined by the points can be uniquely written as aA + bB +cC, where a + b + c = 1. I will give a simple proof that I find elegant. It is taken from Dan Pedoe's book entitled Geometry.
Proof:
If D = A then the theorem is obvious. Assume D != A. Let D' be a point on the line that passes through B and C. Then it is easy to show (write D' = B + t(C-B)) that D' = kB + k'C where k + k' = 1. By the same idea, D = lA + l'D' where l + l' = 1. Then D = lA + l'kB + l'k'C. The sum of the coefficients is l + l'k + l'k' = l + l'(k+k') = l + l' = 1.
The question remains as to whether the coefficients are unique. Assume D = aA + bB + cC = a'A + b'B + c'C, where the sums of both pairs of coefficients is 1. Then (a-a')A + (b-b')B + (c-c')C = 0. The sum of the coefficients in this equation is 1 - 1 = 0. In general, if X, Y, and Z are three non-collinear points and xX + yY +zZ = 0 and x + y + z = 0, then x = y = z = 0. This fact deserves a proof in another writeup but I do not know how to name it. Anyway, that fact proves the uniqueness of the barycentric coordinates.