From: David Iwancio
Date: 2020 Feb 19, 14:46 -0800
If two stars are the same altitude, you are on a great circle that is the perpendicular bisector of the line between the two stars. When the pair are on the horizon, the cut angle is equal to the measured distance between them. When they are overhead, the cut angle is 180° (they are on either side of the zenith). The cut angle determines how far you are from that midpoint. Note that, if the stars are not on the horizon, then the midpoint is higher than the pair.
If the two stars are aligned vertically, you are on the same great circle as the line between the stars. They would be the same "line" of position on your plotting sheet and the cut angle would be 0°. But don't forget that they are actually two circles that actually only "kiss" at a single tangent point; the scale of your plotting sheet is too large to show the different curvatures. However, the line drawn between the stars will cross that point at a right angle.
(If errors give you two circles of position that never cross, however, you end up trying to find a cut angle whose cosine is larger than 1. But you could probably still determine where the lines' closest approach occur.)
I'm still working on a general solution. I'm thinking that finding the altitude of the midpoint may be key.