Homogeneous barycentric coordinates are barycentric coordinates normalized such that they become the actual areas of the subtriangles. Barycentric coordinates normalized so that t_1 + t_2 + t_3 = 1, so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates. Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem.

areal coordinates | barycentric coordinates | trilinear coordinates