---
book: 11
number: 2
id: "XI.2"
kind: "theorem"
uses: ["[[book-11/proposition-1]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.2"
license: "CC-BY-SA-4.0"
---

# XI.2

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

## Proof

For let the two straight lines *AB*, *CD* cut one another at the point *E*; I say that *AB*, *CD* are in one plane, and every triangle is in one plane.

For let points *F*, *G* be taken at random on *EC*, *EB*, let *CB*, *FG* be joined, and let *FH*, *GK* be drawn across; I say first that the triangle *ECB* is 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, a part also of one of the straight lines *EC*, *EB* will be in the plane of reference, and a part in another.

But, if the part *FCBG* of the triangle *ECB* be in the plane of reference, and the rest in another, a part also of both the straight lines *EC*, *EB* will be in the plane of reference and a part in another: which was proved absurd. [[book-11/proposition-1|XI. 1]]

Therefore the triangle *ECB* is in one plane.

But, in whatever plane the triangle *ECB* is, in that plane also is each of the straight lines *EC*, *EB*, and, in whatever plane each of the straight lines *EC*, *EB* is, in that plane are *AB*, *CD* also. [[book-11/proposition-1|XI. 1]]

Therefore the straight lines *AB*, *CD* are in one plane, and every triangle is in one plane. Q. E. D.