I took a bit of a simple route in the attached example. We have the coordinates of the ends of a baseline and azimuths to a third point. No elevations, so just put everything on the ellipsoid by assuming h = 0.
Use INVERS3D to find the mark-to-mark distance and the azimuth from one given point to the other. Use that along with the given azimuths to the unknown point to find the internal angles of a triangle with the two given points and the unknown point as its vertices.
Use the Law of Sines to calculate the distances of the two unknown legs of the triangle. These are mark-to-mark distances.?ÿ
Use FORWRD3D with each mark-to-mark distance and azimuth two calculate the coordinates of the unknown point from each of the given points. The values will be within millimeters of each other.
The key is the closeness of ellipsoidal distances to mark-to-mark distances over short intervals. That allows problems like this to be solved with plane figures, making them much simpler.
The attached PDF shows the work but has no text. All of the NGS output is either to construct the givens or to calculate and evaluate the answer.
Challenging problem, good mix of old school and computers.
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