Let $ABCDEFGH$ be a cube and $I$, $J$, and $K$ be points, where $I$ is the midpoint of edge $AD$, $J$ is the midpoint of edge $FG$ and $K$ is the midpoint of edge $GH$ (see the picture). Construct the cross-section of the cube by the plane $IJK$.
Joey proceeded as follows:
(1) He constructed the line segment $JK$ and claimed that it is one of the sides of the cross-section.
(2) He drew a line $p$ parallel to the line segment $JK$ passing through point $I$. Then he marked the intersection of the line $p$ and edge $AB$ as point $N$. He claimed that the line segment $IN$ is another side of the cross-section.
(3) Joey drew the line segments $IK$ and $NJ$. He claimed that both line segments belong to the cross-section.
(4) He marked the intersection of the line segment $IK$ and edge $DH$ as point $M$ and the intersection of the line segment $NJ$ and edge $BF$ as point $L$. Then he concluded that the hexagon $INLJKM$ is the sought cross-section.
Did Joey make any mistakes? If so, explain.
Yes. There is a mistake in step (1). The line segment $JK$ is not a side of the cross-section.
Yes. There is a mistake in step (2). The line segment $IN$ is not a side of the cross-section.
Yes. There is a mistake in step (3). At least one of line segments $IK$ and $NJ$ does not belong to the cross-section.
Yes. There is a mistake in step (4). The line segment $IK$ and edge $DH$ do not intersect. Similarly, the line segment $NJ$ and edge $BF$ do not intersect.
No. There are no mistakes in the procedure.
There is a mistake in step (4). The line segment $IK$ and edge $DH$ are skew, and so are the line segment $NJ$ and edge $BF$.
The correct procedure is as follows:
(1) Construct the line segment $JK$.
(2) Draw the parallel line $p$ passing through point $I$ and mark the intersection of the line $p$ and edge $AB$ as point $N$.
(3) Extend edge $CB$ and mark the intersection of the line $p$ and the extended edge $CB$ (half-line $q$) as point $L$ (see the picture).
(4) Connect point $L$ with point $J$. Mark the obtained intersection of edge $BF$ and the line segment $JL$ as point $M$.
(5) Draw the line $r$ parallel to the line segment $JM$ passing through point $I$ and mark the intersection of the line $r$ and edge $DH$ as point $O$.
(6) Draw the hexagon $INMJKO$, which is the sought cross-section.
