A rectangular prism has one of its diagonals $A A^{\prime}$. From vertex $A$, we draw a plane through the endpoints of the edges $(B, C, D)$. Into what parts does this plane divide $A A^{\prime}$?
A rectangular prism has vertices $A, B, C, D$ with their opposite vertices $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_752e28b7a1aa4fbaffd8g-1.jpg?height=344&width=420&top_left_y=218&top_left_x=842)

The $A C A^{\prime} C^{\prime}$ face is a rectangle; its diagonals $A A^{\prime}$ and $C C^{\prime}$ bisect each other at point $I$. The plane $B C D$ bisects $A C^{\prime}$ at point $E$ and intersects $A A^{\prime}$ at point $G$.

$C E$ and $A I$ are medians of $\triangle A C C^{\prime}$, intersecting at the centroid $G$ of the triangle, thus

$$
A G=\frac{2}{3} A I=\frac{1}{3} A A^{\prime}
$$

Accordingly, the diagonal starting from vertex $A$ is divided into two parts in a $1: 2$ ratio by the plane passing through the endpoints of the edges starting from vertex $A$.

Lajos Répás (Baross Gábor Gymnasium, 6th grade, Szeged)