From the window of my apartment, I can see two remarkable buildings. I can also spot those two buildings from the place where I work. But I can not spot my apartment from work and I can not spot my working place from my apartment.
Let me refer to the big buildings as tower 1 ($T_1$) and tower 2 ($T_2$), and to my home and working place as $H$ and $W$, respectively.
I have drawn an overview of the situation:
I tried to figure out some of the distances between any two of the four places (There are six lines in total between the four buildings.), or at least the ratio between two of these distances. Just to give you an idea of the situation, my guess is that all six pairwise distances are between 1.5 and 4.5 kilometres.
At first sight, this will not work. I only know two angles, namely $\alpha := \angle T_2 H T_1$ and $\beta :=\angle T_2 W T_1$. I think (am I right?) I have no chance to find out anything about the ratio of the distances from these two angles.
So I became more creative and thought about ways to gain more information. The ground plot of $T_2$ is rectangular, I would even guess it is a square. $T_2$ appears from different perspectives depending on my point of view. By comparing the length of wall segments I see of both visible sides of the tower (and doing this from my apartment as well as from my working place), I can determine the angle $\chi := \angle H T_2 W$. (For a thorough explanation, see footnote [3] below.)
Unfortunately, the building $T_1$ is of circular ground view, so I think I can not apply a similar trick there.
Another idea: I have access to the roof of my working place. (From there, I see the two buildings.) So this is where information about actual (non-relative) distances could enter the picture. However, I can only move maybe 7 metres in one direction before falling off the roof. I suppose this will practically not be enough to get new information, since the inaccuracy when measuring is too high. Right?
I think that knowing $\alpha, \beta, \chi$ is still not enough to gain new information, is it?
So my question is: How could I collect more information about the situation, and how would it help me to determine the distance between any two of the four buildings? Creative ideas welcomed.
[3] How to determine $\chi$, using the shape of tower 2:
This picture
could be helpful for understanding the idea. (Maybe looking at the picture is more helpful than reading the typed explanation.)
When I look in the direction of tower 2 from my apartment, then I see two of its side walls. When I look from my working place, I also see two side walls. In both situations, the prominent side (i.e. the side which "I can see better" or "more") is the same.
Let's focus on the perspective from my apartment, first. Using a somewhat unlucky notation, I denoted as $\alpha$ the angle between the prominent side of tower 2 and the direct connecting line $\overline{T_2 H}$. You can see a sketch in the lower right corner of my uploaded image. From my apartment, the prominent side of tower 2 appears to be $y$ wide; the other side appears to be $x$ wide. Some trigonometry gives $x : y = \cos \alpha : \sin \alpha$. I can measure $x$ and $y$ and then solve this equation for $\alpha \in [0°, 90°]$.
The angle between the non-prominent side of tower 2 and $\overline{T_2 H}$ is just $90° - \alpha$. Proceding similarly, I can determine the angle between the non-prominent side of tower 2 and $\overline{T_2 W}$. Their sum is $\chi$.