We have been waiting for Feri to return from a forest excursion. Finally, his messenger pigeon arrives with the following letter.

"A large branch fell from the lower branch of a tree, and I fell off, breaking my left leg. It's not a big problem. Come and get me. From above, I saw the segment between hills $A$ and $B$ at an angle of $15^{\circ}$, and the segment between hills $B$ and $C$ at an angle of $40^{\circ}$. $B$ was visible between $A$ and $C$. Feri"

From the discussion of Feri's three brothers, we learned that the length of segment $AB$ is $4 \mathrm{~km}$, the length of $BC$ is $8 \mathrm{~km}$, and the length of $AC$ is $10 \mathrm{~km}$; furthermore, that their house is $9 \mathrm{~km}$ from point $C$, $16 \mathrm{~km}$ from point $B$, and $A$ is even farther away. One of them also said: "You search for him in $D$, I'll search in $E$, from there I can bring him home alone."

How far is Feri from their house? We can answer based on a construction and measurement.
We can construct (in a suitable scale) a sketch of the area in question from the data: the mutual positions of points $A, B$, and $C$ from their distances, the $15^{\circ}$ angle subtended by the $i_{1}$, $i_{1}^{\prime}$ arc pair of the $A B$ segment (on both sides of the segment), similarly the $40^{\circ}$ angle subtended by the $i_{2}$, $i_{2}^{\prime}$ arc pair of $B C$, the $9 \mathrm{~km}$ radius circle $k_{1}$ around $C$ and the $16 \mathrm{~km}$ radius circle $k_{2}$ around $B$, and finally the perpendicular bisector $m$ of the $A B$ segment. (It is possible that we get the mirror image of the correct map sketch - a sort of bottom view of the area - but this does not affect the use.)

![](https://cdn.mathpix.com/cropped/2024_05_02_0fd399e58a4b4f04a808g-1.jpg?height=994&width=1136&top_left_y=392&top_left_x=489)

In this case, Feri's accident occurred at one of the two points, $D$ and $E$, where the two arc pairs intersect, such that the line $D B$ or $E B$ separates $A$ and $C$, while the other two intersection points ($G$ and $H$) are on the same side of the line connecting $B$ to these points. (In other words, only those intersection points are relevant that lie on the $15^{\circ}+40^{\circ}=55^{\circ}$ angle subtended by the $i_{3}, i_{3}^{\prime}$ arc pair of the $A C$ segment; from $G$ and $H$, the angle subtended by $A C$ is $40^{\circ}-15^{\circ}=25^{\circ}$.)

On the other hand, both common points of $k_{1}$ and $k_{2}$, $L_{1}$ and $L_{2}$, are possible as Feri's home, because both are on the same side of $m$ as $B$. The information leaves this question ambiguous. According to the measured lengths on the sketch, it is more likely that the family lives in $L_{1}$, which is closer to $m$, because the distance $L_{1} E=13 \mathrm{~km}$ is significantly smaller than $L_{1} D=20 \mathrm{~km}$ - this seems to justify the family's decision that it is easier to transport Feri from $E$ - but for $L_{2}$, which is farther from $m$, the two distances show only a small difference, and in fact $L_{2} E=19 \mathrm{~km}$ is larger than $L_{2} D=17 \mathrm{~km}$. (The area's transportation possibilities could of course modify our ranking.)

Note. Feri's additional data - that he saw $B$ between $A$ and $C$ - can be interpreted as him being aware that the problem of intersection behind the back is not always uniquely solvable given the two angles of view. (His interest in the question is shown by the fact that he made his measurements before the accident.) However, he could have completely resolved the remaining ambiguity by also stating whether he swept his line of sight over $A$, $B$, and $C$ in a clockwise or counterclockwise direction. We could only use this information to unambiguously select between $D$ and $E$ as Feri's location if we also knew that, for example, when circling the three peaks on the map, we perform a positive or negative rotation. - The construction is intended to draw attention to these issues with the problem.

Of course, these doubts disappear if we are standing in the area and have the map in our hand, or if the sketch completes the problem.