We want to determine the position of point $A$ in a mountainous area relative to peaks $B$ and $C$ based on the following data. The horizontal projection of the distance $BC$ is $2400 \, \text{m}$, and peak $B$ is $800 \, \text{m}$ higher than $C$. The angle of elevation of the line of sight $AB$ is $20^\circ$, and that of $AC$ is $2^\circ$. The angle between the directions from $A$ to the horizontal projections $B'$ and $C'$ of $B$ and $C$ on the horizontal plane passing through $A$ is $60^\circ$.

What are the horizontal projections of the distances $AB$ and $AC$, and the height difference between $B$ and $A$?
We denote the sought distances $A B^{\prime}, A C^{\prime}$ as $b$ and $c$ respectively, the height difference $B B^{\prime}$ as $x$, and the given data as follows: $B^{\prime} C^{\prime}=a$, $B C^{\prime \prime}=h$ (where $C^{\prime \prime}$ is the projection of $C$ onto $B B^{\prime}$), $\angle B^{\prime} A B=\beta$, $\angle C^{\prime} A C=\gamma$, and $\angle B^{\prime} A C^{\prime}=\alpha$.

![](https://cdn.mathpix.com/cropped/2024_05_02_c49169be320a7e109dfbg-1.jpg?height=537&width=507&top_left_y=279&top_left_x=793)

Figure 1

We can express $b$ in terms of $x$ and $\beta$, and $c$ in terms of the height difference between $A$ and $C$, $C C^{\prime}$, and $\gamma$. Substituting these into the expression for the side $B^{\prime} C^{\prime}$ of the triangle $A B^{\prime} C^{\prime}$ given by the cosine rule, we obtain a single-variable equation for $x$:

$$
\begin{gathered}
b=x \operatorname{ctg} \beta, \quad c=(x-h) \operatorname{ctg} \gamma \\
x^{2} \operatorname{ctg}^{2} \beta+(x-h)^{2} \operatorname{ctg}^{2} \gamma-2 x(x-h) \operatorname{ctg} \beta \operatorname{ctg} \gamma \cos \alpha=a^{2} \\
x^{2}\left(\operatorname{ctg}^{2} \beta+\operatorname{ctg}^{2} \gamma-2 \operatorname{ctg} \beta \operatorname{ctg} \gamma \cos \alpha\right)- \\
-2 x h \operatorname{ctg} \gamma(\operatorname{ctg} \gamma-\operatorname{ctg} \beta \cos \alpha)+\left(h^{2} \operatorname{ctg}^{2} \gamma-a^{2}\right)=0
\end{gathered}
$$

Using the simpler relationships $2 \cos \alpha=1$ and $a=3 h$ from our data, we transform our equation to use the ratio $x / h=\xi$ as the unknown, and then substitute the given angles $\beta, \gamma$:

$$
\begin{gathered}
\xi^{2}\left(\operatorname{ctg}^{2} \beta+\operatorname{ctg}^{2} \gamma-\operatorname{ctg} \beta \operatorname{ctg} \gamma\right)-\xi\left(2 \operatorname{ctg}^{2} \gamma-\operatorname{ctg} \beta \operatorname{ctg} \gamma\right)+\left(\operatorname{ctg}^{2} \gamma-9\right)=0 \\
789.9 \xi^{2}-1561.4 \xi+811.0=0 \\
\xi^{2}-2.085 \xi+1.083=0 \\
\xi=1.0424 \pm 0.0612
\end{gathered}
$$

According to the problem, $x>h$, so $\xi>1$, and only the larger root is acceptable: $\xi=1.104$, which clearly gives $x=883 \, \text{m}$. From this, using (1), the projections of $A B$ and $A C$ on the map are $2426 \, \text{m}$ and $2374 \, \text{m}$ respectively. (See Figure 2; the dotted line indicates the position of the rejected $C_{2}^{\prime}$, where $B_{2}^{\prime} C_{2}^{\prime}=a$ is satisfied, but in the opposite direction. It is not harmful to consider the possible interpretation of the rejected root.)

![](https://cdn.mathpix.com/cropped/2024_05_02_c49169be320a7e109dfbg-1.jpg?height=550&width=610&top_left_y=1739&top_left_x=736)

Remark. The problem was derived from an engineering question by the editorial team, rounding the data to form the 11-Potenoth problem (or Snellius-Potenoth problem) for resection. The usual resection involves three base points, but here four are used: $B, B^{\prime}, C^{\prime}$, and $C$, and instead of two angles, three are measured at the point $A$ to be determined, two of them vertical and one in the horizontal plane.

The cited article writes down second-degree equations for the sine of the angles at points $B^{\prime}$ and $C^{\prime}$—essentially for the distances $A B^{\prime}$ and $A C^{\prime}$—and then calculates the roots to facilitate the use of the most important computational aid of the time, the logarithm table, by introducing auxiliary angles.[^0]

[^0]: ${ }^{1}$ Zoltán Gyöngyössy: The "spatial Potenoth". Bulletin of the Hungarian Association of Engineers and Architects, 1914, No. 27.

    ${ }^{2}$ See also exercises 1419 and 1424, K. M. L. 44 (1972) 172 and 220.