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Exercise

Given: ABCDEFGH is a cube. The point M is the midpoint of the edge DH. The point N is the center of the face CDHG.

Question: What is the relationship between line AM and BN (parallel, intersecting, or non-coplanar)?

Solution

Line AM and line BN are intersecting lines. The following is a brief explanation, showing that line AM and line BN are on the same plane, and are not parallel to each other. Hence, they must intersect.

Since line AB and line MN are both parallel to the line CD, line AB is parallel to line MN. Therefore, a plane must exist that passes through both lines AB and MN. In fact, this plane also bisects the edge CG at point P. Consequently, points A, B, M, and N are on the same plane. As a result, line AM and line BN are also on the same plane.

Since line AM is parallel to line BP while line BN is not parallel to line BP, line AM and line BN must intersect.

Please follow this link to view their relationship from a variety of angles.


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