Two Intersecting Straight Lines are in One Plane

Theorem

In the words of Euclid:

If two straight lines cut one another, they are in one plane, and every triangle is in one plane.

(The Elements: Book $\text{XI}$: Proposition $2$)

Proof

Let the two straight lines $AB$ and $CD$ intersect at the point $E$.

Let $F$ and $G$ be arbitrary points on $EC$ and $EB$.

Let $CB$ and $FG$ be connected.

Let $FH$ and $GK$ be drawn across.

Suppose part of $\triangle ECB$ is in the plane of reference and the rest of it in another plane.

Then a part of one of the straight lines $EC$ and $EB$ will be in the plane of reference and the result in another plane.

But from Proposition $1$ of Book $\text{XI} $: Straight Line cannot be in Two Planes, this cannot happen.

Therefore $\triangle ECB$ is all in one plane.

But whatever plane $\triangle ECB$ is in, each of the straight lines $EC$ and $EB$ are also in that plane.

And in whatever plane $EC$ and $EB$ are in, the straight lines $AB$ and $CD$ are also in that plane.

Therefore $AB$ and $CD$ are in one plane, and every triangle is in one plane.

$\blacksquare$

Historical Note

This proof is Proposition $2$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions