If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane.

For let the two straight lines *AB* and *CD* cut one another at the point *E.*

I say that *AB* and *CD* lie in one plane, and that every triangle lies in one plane.

Take the points *F* and *G* at random on *EC* and *EB,* join *CB* and *FG,* and draw *FH* and *GK* across.

I say first that the triangle *ECB* lies in one plane.

For, if part of the triangle *ECB,* either *FHC* or *GBK,* is in the plane of reference, and the rest in another, then a part also of one of the straight lines *EC* or *EB* is in the plane of reference, and a part in another.

But, if the part *FCBG* of the triangle *ECB* is in the plane of reference, and the rest in another, then a part also of both the straight lines *EC* and *EB* is in the plane of reference and a part in another, which was proved absurd.

Therefore the triangle *ECB* lies in one plane.

But, in whatever plane the triangle *ECB* lies, each of the straight lines *EC* and *EB* also lies, and in whatever plane each of the straight lines *EC* and *EB* lies, *AB* and *CD* also lie.

Therefore the straight lines *AB* and *CD* lie in one plane; and every triangle lies in one plane.

Therefore, *if two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane. *

Q.E.D.

Postulates of some sort are needed to justify the existence of planes. One could state that three noncollinear points determine a plane. Another might be that there are four noncoplanar points.