I got the following definition of what an angle between the points A B C in the 2 dimensional euclidean plane is. $$angle(A, B, C) := arccos ((A-B, C-B) / (||A-B|| * ||C-B||))$$
For this definition I want to formally prove that:
lemma: If $P \in convex hull \{A, B, C\}"$ then $$angle(A, B, C) = angle(A, B, P) + angle(P, B, C)$$
I know that I can translate my whole situation, so that B is the origin to obtain a simpler equation. I can even rotate the situation, so that one of A, C or P have one zero component. Then I tried to look at a convex combination of P, but I have not succeeded to find an answer. Have I overseen something, how would a valid proof of this lemma look like. Thanks in advance.
If one understands that $$\operatorname {angle}(A, B, C) := \operatorname{arccos} \left(\frac{(A-B, C-B) } {||A-B|| \cdot ||C-B||}\right)$$ computes the angle in blue shown in figure (a) below
then the one will understand how to compute the red angles shown in figures (b) and (c).
The sum of the red angles equals the blue one.