I want to show that if I have two Euclidian vectors in $\mathbb{R}^n$ than the sum of these two vectors bisects the angle between the two vectors. Said more mathematically.
Let $(u,v) \in \mathbb{R}^n$
Then $\angle(u,v+u) = \angle(u+v,v)$ if and only if $|u|=|v|$
I tried using the fact that
$ \angle(u,v) = \arccos \left( \frac{u \cdot v}{|u| |v|} \right) $
Alas this attempt was futile
Now for $\mathbb{R}^2$ this is obviously true, as can bee seen from my illustration.
How do I prove this with some rigor? Thanks for all tips and advices, this is not a homework question.