\section*{Problem 2 - 101042}

From a cuboid with vertices \(A, B, C, D, A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}\) and edge lengths \(A B = a\), \(A D = b\), \(A A^{\prime} = c\), parts are cut off using plane sections through the vertices \(B^{\prime}, A, D^{\prime}\) or \(A^{\prime}, B, C^{\prime}\) or \(A^{\prime}, D, C^{\prime}\) or \(B^{\prime}, C, D^{\prime}\), which contain the vertices \(A^{\prime}\) or \(B^{\prime}\) or \(C^{\prime}\) or \(D^{\prime}\) respectively. The volume of the remaining body is \(V_{R}\), and that of the original cuboid is \(V_{Q}\).

a) List all points of the cuboid that are vertices of the remaining body, and draw this in an oblique view with \(\alpha=60^{\circ}, q=1\).

The oblique view is to be drawn for the case \(a=5 \mathrm{~cm}, b=2 \mathrm{~cm}, c=2.5 \mathrm{~cm}\).

b) Calculate \(V_{R}: V_{Q}\).
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![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1292.jpg?height=394&width=694&top_left_y=2253&top_left_x=293)

Let the four intersecting planes be denoted by \(E_{1}, E_{2}, E_{3}, E_{4}\) in the order specified in the problem statement.

\(E_{1}\) and \(E_{2}\) intersect the top face of the cuboid along the diagonals \(B^{\prime} D^{\prime}\) and \(A^{\prime} C^{\prime}\), respectively. The intersection point \(S\) of these diagonals is common to the planes \(E_{1}\) and \(E_{2}\).

\(E_{1}\) and \(E_{2}\) intersect the side face \(A B B^{\prime} A^{\prime}\) of the cuboid along the diagonals \(A B^{\prime}\) and \(A^{\prime} B\), respectively.

The intersection point \(P_{12}\) of these diagonals is common to the planes \(E_{1}\) and \(E_{2}\). Consequently, the line \(S P_{12}\) is the intersection line of the planes \(E_{1}\) and \(E_{2}\).

Due to the convexity of the cuboid, the intersection line has exactly the segment \(S P_{12}\) in common with the cuboid.

By similar reasoning, the line \(A P_{12}\) is the intersection line of \(E_{1}\) with the plane spanned by the points \(A B B^{\prime} A^{\prime}\). Similarly, \(B P_{12}\) lies in the intersection line of \(E_{2}\) and \(A B B^{\prime} A^{\prime}\).

The points \(A, B\), and \(P_{12}\) lie below the planes \(E_{3}\) and \(E_{4}\) along with the rest of the body. It follows that the segments \(A P_{12}, B P_{12}, S P_{12}\) are edges of the remaining body. The edge \(A B\) remains unchanged by these four cuts.

By cyclic continuation of these considerations, it is found that the points \(A, B, C, D\) remain as vertices of the remaining body, while the vertices \(A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}\) are lost. Exactly the midpoints \(P_{12}, P_{23}, P_{34}, P_{41}\), and \(S\) of the side faces and the top face are added as new vertices.

A convex remaining body with 9 vertices, 16 edges, and 9 faces remains. The check using Euler's polyhedron formula \(f + v - e = 2\) is satisfied.

The volume \(V_{R}\) of the remaining body can be related to the volume \(V_{Q}\) of the cuboid as follows:

Lay two side-parallel plane cuts through the point \(S\), which divide the cuboid into four congruent cuboids. By applying the four plane cuts, for example, the part-cuboid with the edge \(A A^{\prime}\) is divided into two equal-volume sub-bodies by the plane \(E_{1}\), with exactly the lower part falling to the remaining body.

Applying this reasoning to all four part-cuboids results in the statement \(V_{R}: V_{Q} = 1: 2\).

Adapted from \([5]\)