Angle Bisector Vector/Geometric Proof 2

Proof
The vectors $\norm {\mathbf u} \mathbf v$ and $\norm {\mathbf v} \mathbf u$ have equal length from Vector Times Magnitude Same Length As Magnitude Times Vector.

Thus $\norm {\mathbf u} \mathbf v + \norm {\mathbf v} \mathbf u$ is the diagonal of a rhombus.

The result follows from Diagonals of Rhombus Bisect Angles.