How is distance between two points defined in barycentric coordinates?

Question

Hope someone can help. I have this 3-d simplex (a tetrahedron) and its vertexes have barycentric coordinates defined as follow: $A_1=(1,0,0,0), A_2=(0,1,0,0), A_3=(0,0,1,0), A_4=(0,0,0,1)$. I am wondering what is the distance between, for example, $A_1$ and $A_2$? I know that an any point T on the edge $\bar{A_1A_2}$ is determined by a number t $\in$ [0,1] such that $\frac{t}{1-t}=\frac{\bar{A_2T}}{\bar{TA_1}}$ and its barycentric coordinates thus are (t,1-t,0,0). I intuitively understand it, but why is it like this? Thank you in advance