A right triangle ABC has the angle $\hat{C} = 90^\circ$ and $\hat{A} = \theta \in (0, 90^\circ )$. The length of the side $AC = b$. Consider $C_1$ to be the point on $AB$ such that $CC_1$ is perpendicular to $AB$, $C_2$ the point on $BC$ such that $C_1C_2$ is perpendicular to $BC$, $C_3$ the point on $AB$ such that $C_2C_3$ is perpendicular to $AB$, and so on (meaning, this process continues indefinitely). Show that the total length of all the perpendiculars
$CC_1 + C_1C_2 + C_2C_3 + \dots$
is finite and find its exact value in terms of $b$ and $\theta$.
I've tried drawing a picture and my hint is to compute $CC_1$ but I don't know how to do that and I'm kind of confused on what some of these terms are even referring to. Any help is appreciated.
Note that we have a geometric series. $$cc_1 =b\sin \theta $$
$$c_1c_2 = cc_1 \sin \theta= b\sin^2 \theta $$
$$c_2c_3 = c_1c_2 \sin \theta= b\sin^3 \theta $$
Thus the total sum is $$S= b \sin \theta +b\sin^2 \theta + b\sin^3 \theta+... $$
Which is convergent with the sum
$$ S = \frac {b\sin \theta}{1-\sin \theta} $$