# Determine arc angles of line intersecting three concentric circles

I have been wracking my brain on this problem and cannot derive the equation. It is related to tooth profiles on spur gears.

Given three equidistant concentric circles, with the inner circle radius r, the middle circle radius r+d, and the outer radius r+2d. A line intersects all three circles such that the angle formed by an intersecting radius-line with the middle circle is θ.

What are the resulting arc-angles α¹ and α² that result from the line intersecting the inner and outer circles? I actually need the sum of the angles, so if it is easier to solve as a single arc, even better. Thanks!

image describing the geometry

If I got your question right, you just need to apply sine theorem to the triangles formed by (1) radius $r+2d$ radius $r+d$ and the intersecting line, and (2) radius $r+d$ radius $r$ and the intersecting line. The angles of triangle (1) are $\alpha_2$, $\,\,\theta - \alpha_2$ and $\pi - \theta$ and (2) $\alpha_1$, $\,\,\theta$ and $\pi - (\alpha_1 + \theta)$. Hence $$\frac{r+2d}{\sin(\theta)} = \frac{r+d}{\sin(\theta - \alpha_2)}$$ $$\frac{r+d}{\sin(\alpha_1+\theta)} = \frac{r}{\sin(\theta)}$$ Solve for $\alpha_2$ and $\alpha_1$