---
book: 11
number: 17
id: "XI.17"
kind: "theorem"
uses: ["[[book-11/proposition-16]]", "[[book-6/proposition-2]]", "[[book-5/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.17"
license: "CC-BY-SA-4.0"
---

# XI.17

*If two straight lines be cut by parallel planes*, *they will be cut in the same ratios.*

## Proof

For let the two straight lines *AB*, *CD* be cut by the parallel planes *GH*, *KL*, *MN* at the points *A*, *E*, *B* and *C*, *F*, *D*; I say that, as the straight line *AE* is to *EB*, so is *CF* to *FD*.

For let *AC*, *BD*, *AD* be joined, let *AD* meet the plane *KL* at the point *O*, and let *EO*, *OF* be joined.

Now, since the two parallel planes *KL*, *MN* are cut by the plane *EBDO*, their common sections *EO*, *BD* are parallel. [[book-11/proposition-16|XI. 16]]

For the same reason, since the two parallel planes *GH*, *KL* are cut by the plane *AOFC*, their common sections *AC*, *OF* are parallel. [*id*.]

And, since the straight line *EO* has been drawn parallel to *BD*, one of the sides of the triangle *ABD*, therefore, proportionally, as *AE* is to *EB*, so is *AO* to *OD*. [[book-6/proposition-2|VI. 2]]

Again, since the straight line *OF* has been drawn parallel to *AC*, one of the sides of the triangle *ADC*, proportionally, as *AO* is to *OD*, so is *CF* to *FD*. [*id*.]

But it was also proved that, as *AO* is to *OD*, so is *AE* to *EB*; therefore also, as *AE* is to *EB*, so is *CF* to *FD*. [[book-5/proposition-11|V. 11]]

Therefore etc. Q. E. D.