Relation between product of reflections and angle

Question

I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras.

It's known that Weyl Group $W$ is generated by set of reflections.

If I consider an arbitrary element of Weyl group $\sigma_\alpha \sigma_\beta$,($\alpha,\beta \in \Phi$) then it's easy to calculate it's order by definition of reflection if angle between $\alpha$ and $\beta$ is $\pi/2$.

But how to see the fact that $\sigma_\alpha \sigma_\beta$= rotation through $2\theta$( if $\theta$ is angle between $\alpha$ and $\beta$). I can't visualise it.

Please see!

Answer 1

This drawing in the two-dimensional case should illuminate things.