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I found the following statement:

Let $A, C$ and $E$ be three distinct points on the line $l_1$ and $B,D,F$ three distinct points on the line $l_2$. Let us assume that $AB\cap DE=L$, $CD\cap FA=M$, and $EF\cap BC=N$. Then $L,M,N$ are collinear.

In the picture of the paper I'm reading, it looks like $C$ lies between $A$ and $E$ and $D$ lies between $B$ and $F$. Do we need this assumption or does the theorem work no matter in what order the points are on the line? And how do I prove the case that some of the lines $AB$, $CD$ and $EF$ are parallel?

guest
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1 Answers1

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The order matters, because the lines (e.g. $AB$, $DE$, $EF$, $BC$, etc.) with one point on $l_1$ and one on $l_2$ could be parallel, and so $L$, $M$, or $N$ might not exist.

For example, if the points appeared in the order $A$, $E$, $C$ on $l_1$ and $B$, $D$, $F$ on $l_2$, then we could easily have $$AB \cap DE = \varnothing$$

Zubin Mukerjee
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