A ,B,C,D are four points each four different planes. If we draw all the lines between the points , can some or all of them be parallel with each other?
2026-03-25 20:59:54.1774472394
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Can this line be parallel with each other?
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- Every set of 3 non-collinear points define a plane, and if there is a set of 3 co-linear points then those 3 points and the fourth define a plane (unless those 4 points are collinear that is, then there are many planes containing all 4 points). So what you (probably) mean is
no set of 3 co-linear points, and every point not coplanar with the plane specified by the other 3 points.
- Assuming the shaded: There are 6 lines specified here: the lines defined by $(A,B), (A,C), (A,D),(B,C), B,D)$, and $(C,D)$. Each line necessarily intersects with 4 other lines e.g., $(A,B)$ intersects with the lines specified $(A,C),(A,D), (B,C), (B,D)$.
However, the line specified by $(A,B)$ is necessarily parallel to the line specified by $(C,D)$. Why?
Fact 1: Let $P$ be a plane and let $L$ be a line. If $L$ is not a line in $P$ then $L$ intersects $P$ at most once.
By Fact 1, the line specified by $(C,D)$ [as $D$ is not on the plane specified by $A,B,C$] intersects the plane specified by $A,B,C$ at exactly one point, namely $C$. However, the line defined by $(A,B)$ has every point in the plane specified by $A,B,C$ so if it were to intersect the line specified by $(C,D)$, it would do so at $C$. However, the line specified by $(A,B)$ does not contain $C$ because no 3 points are collinear.
Likewise the line specified by $(A,D)$ is necessarily parallel to the line specified by $(B,C)$, and the line specified by $(A,C)$ is necessarily parallel to the line specified by $(B,D)$.
The argument breaks down if there is a plane containing all 4 points, in fact if $A,B,C,D$ are 4 randomly chosen points on a plane then the line specified $(A,B)$ interscts the line specified by $(C,D)$, and likewise for the other lines.
It is possible to project 4 parallel lines from the 4 points on the first plane to each of the other 3 planes. With no restriction on the position of the points, this is easy to visualize.