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Thanks to both of you Lars and Bill for your kind replies.

Bill, I just downloaded I. Todhunter's Spherical Trigonometry Book. It looks great and I will use it until I get my trusty "Astronomie Générale" by André Danjon out of its cardboard storage box.

Pending further check from André Danjon's book which still remains a goldmine - to me at least - and as per Todhunter's Book Articles 84 and 86, it looks like there is definitely no unambiguous solution to solving a spherical triangle from 2 sides and 1 non-adjacent angle .

In the iterative method given here, there is no such unambiguity, but I highly prefer direct solutions if and when any.

As Lars noticed it, for this Problem submitted by Dave Walden Azimuth solving is irrelevant since we even know the Azimuth. Moreover, and given the relative positions of Sun and Observer, there is even little ambiguity if ever that we are to choose the value under 90° for the LHA.

However ... there still remains an ambiguity in the general case, and if we were to program a solution to this problem, we certainly better deliver a software tackling all sorts of possible situations.

One method could be to first retain the LHA smaller than 90° and to carry on. Once all solved, compute the height from the derived Observer's Position and check for consistency with the initially published Height. If check unsatisfactory, repeat the computations with the LHA bigger than 90°. This one should work, hopefully ...

Peter, or Andrés, how did you tackle this possible LHA indetermination trap in your spreadsheets?

And once we get replies from our dear colleagues, I think this story becomes more or less over.

Dave, Many Thanks again for submitting such an earlier days refreshing exercise. 

Best Regards to all,

Antoine