# Problem 2.

In the Kingdom of Magic Energy, an enchanted transformer booth stands on a flat plain: to an observer looking parallel to the ground, it is visible only at an angle of $90^{\circ}$. 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 is visible. 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.

It is enough to recall that any inscribed angle in a circle that subtends the diameter is a right angle. Therefore, the desired set will be four semicircles (with diameter $L$) subtending the sides of the square (see the figure below). For clarity, one of the right angles from which the tower is visible from the point $Q$ is also shown.

![](https://cdn.mathpix.com/cropped/2024_05_06_28a3194dd343bf33cb92g-2.jpg?height=571&width=854&top_left_y=842&top_left_x=544)

Obviously, the minimum distance is 0 (from the corner of the square), and the maximum distance is $L / 2$ (from the farthest point on the circle to the midpoint of the side of the square).

Answer: Four semicircles subtending the sides of the square (see the dotted line in the figure above); $\quad \rho_{\min }=0 ; \quad \rho_{\max }=L / 2$.