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As you indicated, knowing three sides of the spherical triangle - in this case, the observed lunar distance (OLD), and HO’s of the sun and moon - completely determine the triangle and allow one to calculate the “LHA” (actually difference in azimuths of the sun and moon). From there, after determining the HC’s of the bodies by adjusting for refraction and parallax, one can calculate the geocentric lunar distance (LD). The process involves no mathematical approximations, and the only inaccuracies come from measurements of the HO’s and OLD, and from adjusting for refraction and parallax. But this process is computationally dense, and there are other ways to clear the lunar distance. 

However, I drew this diagram not to clear the lunar distance, but to show how to calculate the OLD at an assumed time from a corresponding assumed longitude and defined latitude. That calculation is necessary to get the GMT and longitude from a lunar distance measurement by the iterative calculation methods described in more detail at https://www.starpath.com/resources2/brunner-lunars.pdf . That iterative process is also summarized in another post. The azmuths and HC’s of the sun and moon are calculated from the assumed time and position. The “LHA” is calculated by determining the difference of the azimuths, and the HO’s are calculated by subtracting the parallaxes (you can probably skip the parallax of the sun) and then adding the refraction corrections to the HC’s. From the HO’s and “LHA”, the Observed Lunar Distance (OLD) is calculated, after subtracting the semi-diameters including augmentation. This is essentially the inverse of the more familiar “clearing the lunar distance”, where the geocentric LD is calculated from the OLD.

The parallax correction is P = HPcos(HO). However, we don’t know HO; that is what we are trying to calculate from HC. In my article I used P = HPcos(HC) as an approximation, but that was a mistake and is not accurate enough. A better approximation is P= HPcos(HC-HPcos(HC)); that turns out to be accurate enough for lunar work.  

As well as for working lunars, calculating the OLD at a known time and position (like your back yard) can give you feedback when practicing lunar distance observations. Again, the calculations are a little intense. Maybe nowadays there is some website that will give you OLD from a known time and position; my calculator has a programmed routine for this.