Sun, Moon: Relative Distance from Crescent? This is more a question about the history of astronomy.

It recently occurred to me that you can estimate the ratio of earth-moon and earth-sun distance in the following way: We can look at the triangle observer-moon-sun. Call the angle at the observer $\alpha$, the angle at the moon $\beta$.


At the crescent to half moon stage, sun and moon are both visible, so one can determine the angle $\alpha$ (at the observer).
From the illuminated fraction of the moon, one can estimate $\beta$. (Precision is presumably lower here.)
Thus, the third angle is fixed, and the shape of the triangle is known (but not the overall scale).


So it should be possible to derive that the sun is much further away than the moon. If we could get an independent determination of, say,  the earth-moon distance, the distance to the sun would be known.

This seems rather straightforward. Has this been used historically? What accuracy was (could have been) achieved? For example, I seem to recall that in the "celestial spheres" approach, people have at least ordered the heavenly bodies according to distance, but has this method been taken into account?
This is exactly how the ancient Greeks attempted to estimate the relative distances to the sun and moon.

The difficulty is the angles involved are rather small

Aristarchus measured the Moon-Earth-Sun angle at half moon to be 87 degrees. (or 3 degrees less than a right angle) Which implies the Sun is about 19 times further than the moon. And from this (and an estimate of the sizes of the Earth and moon made at a lunar eclipe) he estimated the distance of the sun.

The actual angle is more like 89.85 degrees (which is very hard to distinguish from 90 degrees) (0.15 degrees less than a right angle). 

So this method can be done, but it requires very exact measurement to succeed.