# Problem 3.

In the Kingdom of Enchanted Energy, an enchanted transformer booth stands on a flat plain: an observer looking parallel to the ground can see it only at an angle of $45^{\circ}$. In cross-section, the booth is square with a side length of $L$ cubits. Describe the geometric locus of points on the plain from which the booth is visible, and determine the minimum and maximum distances from which the enchanted booth can be seen. The angle at which a figure $F$ is visible from a point $P$ is defined as the smallest angle with vertex $P$ that contains the figure $F$. In this case, this angle is located in the plane of the cross-section of the booth.
# Solution.

As is known, any inscribed angle in a circle is half the measure of the central angle that subtends the same arc.

Therefore, an arbitrary segment $A B$ will be seen at an angle of $45^{\circ}$ if the vertex of this angle $Q_{i}$ is located on the circle with the center at the vertex of a right isosceles triangle $O$, whose hypotenuse coincides with the given segment $A B$ (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_c10da7d2de043162f016g-3.jpg?height=794&width=785&top_left_y=1236&top_left_x=744)

We apply this established fact to consider a square. It is clear that there are points from which only one side of the square will be visible, and points from which two adjacent sides will be visible. Other situations are impossible.

In the first case, the segment $A B$ will be the side of the square. The center of the arc of the circle, which is the desired locus of points, will lie at the vertex (right angle) of an isosceles right triangle constructed on this side (see figure above). The maximum distance from this arc to the side of the square is marked on the figure with a dashed line. It is equal to

$$
O H+O F=L / 2+L / \sqrt{2}=\frac{1+\sqrt{2}}{2} L
$$

In the second case, the segment $A B$ will be the diagonal of the square. The center of the arc of the circle, which is the desired locus of points, will lie at the vertex of the original square where the visible sides meet. The distance from any point on this arc to the vertex of the square is $L$. This will be the minimum of the desired distances.

Together, this gives eight quarter-circles, as shown below.

Answer: The desired arcs of circles are shown below with solid lines. Dashed lines mark the lines separating the eight zones.

$$
\rho_{\min }=L, \quad \rho_{\max }=\frac{1+\sqrt{2}}{2} L
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_c10da7d2de043162f016g-4.jpg?height=914&width=905&top_left_y=1096&top_left_x=470)