Geometrically finding the axis of the composition of two rotations

Artin, in his Algebra, has an exercise which states

Find a geometric way to determine the axis of rotation of the composition of two three-dimensional rotations.

I have a possible solution, which I have no idea how to prove. Here is a diagram roughly showing what I think works:

1. Draw circles of arbitrary size around the axes of rotation.

2. Take the point $P$ on the first circle that, when rotated by $\theta_1$ (the green angle) clockwise about $\Gamma_1$, is mapped onto the closer of the two points of intersection of the two circles. This point of intersection is now rotated by $\theta_2$ (the blue arc) about $\Gamma_2$ to get the final image point $Q$.

3. Construct a circle $\xi$ tangent to the first circle at $P$ and the second circle at $Q$.
4. The line joining the centre of $\xi$ to the origin is the axis of rotation, and the angle of the composed rotation is the one subtended by the orange arc.

Does this work? Why (not)?

(I have just seen this question about the same exercise, but I have not read any of the answers yet.)