Barycentric coordinates are "weights" (u, v, w) that determine a point P on a triangle (A, B, C) by weighting the triangle corner points. You can calculate the cartesian coordinates of P = u * A + v * B + w * C.
Since the weights u+v+w = 1 you actually don't need all three: u = 1-v-w, so P = (1-v-w) * A + v * B + w * C = A + v * (B - A) + w * (C - A).
Since the weights u+v+w = 1 you actually don't need all three: u = 1-v-w, so P = (1-v-w) * A + v * B + w * C = A + v * (B - A) + w * (C - A).