A given rectangular prism has a base $A B C D$, and the midpoint line of its top face parallel to $A B$ is $E F$, with the midpoint of this segment being $G$. The point $X$ is on the segment $E G$, and the reflection of $X$ over $G$ is $Y$. In which position of point $X$ will the sum $A X+D X+X Y+$ $Y B+Y C$ be the smallest?
Based on the symmetries of the rectangular parallelepiped, the sum in question has four equal terms, so it is sufficient to examine the variation of the function \( f(x) = G X + 2 X A \) (half of the sum) as \( G X = x \) varies in the interval \([0, G E]\). Let \( A B = 2 b \) and \( A E = e \), thus

\[
f(x) = x + 2 \sqrt{(b-x)^{2} + e^{2}}
\]

The condition for the derivative to vanish is

\[
f^{\prime}(x) = 1 - \frac{2(b-x)}{\sqrt{(b-x)^{2} + e^{2}}} = 1 - \frac{2 \cdot X E}{X A} = 0
\]

which implies

\[
\frac{X E}{X A} = \cos \angle E X A = \frac{1}{2}, \quad \angle E X A = 60^{\circ}
\]

This holds if and only if, returning to the use of variables and considering the specified domain of interpretation,

\[
\frac{b-x}{\sqrt{(b-x)^{2} + e^{2}}} = \frac{1}{2}, \quad x = b - \frac{e}{\sqrt{3}} \geq 0, \quad b \geq \frac{e}{\sqrt{3}}
\]

In other words, if in the plane \( G E A \) a line that forms a \( 30^{\circ} \) angle with \( A E \) intersects the midpoint line of the cover at point \( X \) between \( E \) and \( G \), then our function can have an extremum only in this case, or precisely at \( G \).

![](https://cdn.mathpix.com/cropped/2024_05_02_065ea68fe4069cc7c2f3g-1.jpg?height=864&width=1022&top_left_y=972&top_left_x=541)

In any case, \( f^{\prime}(b) = 1 > 0 \). If \( X \) is in the interior of segment \( G E \), then \( \angle E G A < \angle E X A = 60^{\circ} \), and thus

\[
f^{\prime}(0) = 1 - \frac{2 b}{\sqrt{b^{2} + e^{2}}} = 1 - 2 \frac{G E}{G A} = 1 - 2 \cos \angle E G A < 0,
\]

so \( f^{\prime}(x) \) is increasing in the interval, transitioning from negative to positive, and thus \( f(x) \) has a minimum at the found point. If the point where \( f^{\prime}(x) = 0 \) is such that \( x < 0 \), and including \( x = 0 \), then \( f(x) \) is increasing in the interval \([0, G E]\), so its minimum is at \( x = 0 \), when \( X \) coincides with \( G \).

Remarks. 1. The problem of finding the minimum of \( 2 \cdot A X + X G \) in a plane, if instead of \( D \) we take the reflection \( D^{\prime} \) of the cover plane, and do not initially restrict \( X \) to the line \( E F \) but only to the plane of triangle \( A G D^{\prime} \). It is known that in this case the sum \( X A + X G + X D^{\prime} \) is minimized at the isogonal (or Torricelli) point of the triangle, where the viewing angles \( \angle A X G, \angle G X D^{\prime} \), and \( \angle D^{\prime} X A \) are all \( 120^{\circ} \). This point exists only if no angle of the triangle is greater than \( 120^{\circ} \); if there is such an angle, the sum is minimized at the vertex of that angle, and the viewing angles are not equal. - In our case, since \( G A = G D^{\prime} \), such an angle can only be at \( G \), which the solution excluded (highlighted). - The isogonal (or Torricelli) point of triangle \( A G D^{\prime} \) lies on the altitude \( E G \) due to the symmetry of the triangle.

2. Returning to the original problem, our question is related to the problem of finding the minimum total length of a network connecting the points \( A, B, C^{\prime}, D^{\prime} \) (in a plane) (see Problem P.20, K. M. L. 39 (1969) p. 215, and a special case in Exercise Gy. 1317, 44 (1972) p. 112).
3. It is easy to conjecture that if we do not find a point \( X \) (and \( Y \)) on the midline \( E F \), then the appropriate point will be on the midline of the cover parallel to \( A D \).
4. The problem can be considered a part of the following (simplified) practical problem: two perpendicular channels of different widths divide a city into 4 parts. We want to build a pedestrian bridge over the intersection with staircases, which will establish connections between any two parts of the city. (The height is determined by navigation. In this case, the slope of the stairs must also be considered.)