Beta is the proportion of the area of the triangle represented by (Point, A, C). This triangle as well as the other triangles related to alpha and gamma have to make up the entire triangle, so the proportions of alpha, beta, and gamma have to add up to 1.
I also think that if any of alpha, beta or gamma would be negetive, but they add up to one. Then the point will be outside the triangle.
yup, just because they add to 1 doesn't necessarily mean that they will be in the triangle, you also need to verify their values are between 0 & 1