$L_1$ and $L_2$ are two rays drawn from A at an angle 30. A point B is taken at $L_1$ at a unit distance from A .

Question

A perpendicular $BB_1$ is drawn rom B to $L_2$ and perpendicular $B_1B_2$ is drawn from $B_1$ to AB. Perpendicular $B_2B_3$ is drawn from $B_2$ to $AB_1$ and so on. Prove that $AB, AB_1 , AB_2$ are in a geometric progression.

Here is the diagram I managed to draw

I could apply the sine rule for $AB_1$ and $AB$ but I can’t solve for $AB_2$ .what should I do?

Answer 1

Look at triangle $AB_1B_2$. Now $\cos 30º = \frac{AB_2}{AB_1}$.