So let us speak of plane geometry only.
I don't know if we can treat degrees like any ordinary units. A degree is not a "solid" unit, if you know what I mean.
For the purpose of your post here, let me consider degree as an ordinary "solid" unit. Then you might need to transform my explanations into spherical units of measures.
In your example in the diagram, when you moved to your new location X(10deg, 0deg), the original C(20deg, 180deg) became C((10deg +20deg), (0deg +180deg)) = C(30deg, 180deg). My understanding there is that you altered the original (alt, az) by adding alts and azs as if the degrees were "solid" units.
So, following that procedure, the new location of point B now would be:
Draw a straight line segment connecting X and B.
>>>For the new altitude, we will use the Pythagorean theorem,
(XB)^2 = (OX)^2 +(OB)^2
XB = sqrt[(10deg)^2 +(20deg)^2] = 22.36068 deg ------new alt of B.
>>>For the new azimuth of B, we find the angle that OB makes with say, OX.
tan(angle OXB) = (OB)/(OX) = (20deg)/(10deg) = 2
So, angle OXB = arctan(2) = 63.43495 deg
And so, new azimuth = 180deg -63.43495deg = 116.56505 deg
Therefore, new B would be B(22.36068deg, 116.56505deg) ----***
Following same way, the new D would be
D(22.36068deg, 243.43495deg) ---------------***
If my assumption above were correct, then the altitudes of some points would exceed 90 degrees because the hypotenuse is always longer than any leg of the right triangle.
If that is not allowed, then the altitude of the new B should not be governed by the hypotenuse XB. Maybe the altitude of the new B is still 20 degrees. Thus new B is B(20deg, 116.56505deg)?
And new D is (20deg, 243.43495deg)?