How to calculate the red lines on the following triangle?

Question

How can I calculate the length of the red lines that are marked on this triangle

All the information I have is that $a$ = 1 m and that $\alpha=60^\circ$.

Answer 1

First you have $b_1=a \sin(\alpha)$
Then you can see that the right angle triangle are similar, with $\alpha$ as an angle. That yields: $\frac{b_{n+1}}{b_{n} \cos(\alpha)}=\cos(\alpha)$ and therefore $b_{n+1}=b_n \cos^2(\alpha)$
So $b_2=a \sin(\alpha) \cos^2(\alpha)$
So $b_3=a \sin(\alpha) \cos^4(\alpha)$
So $b_4=a \sin(\alpha) \cos^6(\alpha)$
(assuming the . angle is 90 degrees)
And more generally $b_n=a \sin(\alpha) \cos^{2(n-1)}(\alpha)$