A previous slide (slide 21) says that $\alpha + \beta + \gamma = 1$. Here it says $\alpha$ is a ratio between the distance from a point to BC and the distance from A to BC. Why does $\alpha + \beta + \gamma = 1$? Is there a theorem about it?

I suppose if a point is exactly on a vertex, ie $A$, then a would be $1$ (since $1 - a$ has to be $0$) and both $b+c$ would be $0$? Not sure about a definitive proof for inside the triangle though, am curious about this as well.

Btw I get a server error when I try to type alpha/beta/gamma, how do you do it?

For the question above:

I think this slide just takes \alpha + \beta + \gamma = 1 as ground truth and this \alpha is exactly the same as previous slide.

Here is my intuitive proof of \alpha + \beta + \gamma = 1: Think about the system as a physics system with three masses, with weights ma, mb, mc and locations Va, Vb, Vc. Suppose ma + mb + mc = M. Then, the location of center of mass should be V = maVa/M + mbVb/M + mc*Vc/M. (First, calculate the center of mass of (B and C), then calculate the center of mass of A and (B and C), easy to prove)

Obviously, ma/M + mb/M + mc/M = 1. We can assign the center of mass to any location we want via changing the weight of three masses. Hence, it can be extended to the Barycentric Coordinates with \alpha = ma/M, \beta = mb/M, \gamma = mc/M

@saltyminty -- I edited your comment lightly to make your variables show up correctly.

Take a look at our Markdown article to see how to do it -- very easy and recommended!

@LeslieTrue, same comment for you, and you should be able to edit your own comment to test out the Markdown on your math.

And: good technical discussion!