---
book: 11
number: 27
id: "XI.27"
kind: "construction"
uses: ["[[book-6/proposition-12]]", "[[book-5/proposition-22]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:11.prop.27"
license: "CC-BY-SA-4.0"
---

# XI.27

*On a given straight line to describe a parallelepipedal solid similar and similarly situated to a given parallelepipedal solid.*

## Proof

Let *AB* be the given straight line and *CD* the given parallelepipedal solid; thus it is required to describe on the given straight line *AB* a parallelepipedal solid similar and similarly situated to the given parallelepipedal solid *CD*.

For on the straight line *AB* and at the point *A* on it let the solid angle, contained by the angles *BAH*, *HAK*, *KAB*, be constructed equal to the solid angle at *C*, so that the angle *BAH* is equal to the angle *ECF*, the angle *BAK* equal to the angle *ECG*, and the angle *KAH* to the angle *GCF*; and let it be contrived that, as *EC* is to *CG*, so is *BA* to *AK*, and, as *GC* is to *CF*, so is *KA* to *AH*. [[book-6/proposition-12|VI. 12]]

Therefore also, ex aequali, as *EC* is to *CF*, so is *BA* to *AH*. [[book-5/proposition-22|V. 22]]

Let the parallelogram *HB* and the solid *AL* be completed.

Now since, as *EC* is to *CG*, so is *BA* to *AK*, and the sides about the equal angles *ECG*, *BAK* are thus proportional, therefore the parallelogram *GE* is similar to the parallelogram *KB*.

For the same reason the parallelogram *KH* is also similar to the parallelogram *GF*, and further *FE* to *HB*; therefore three parallelograms of the solid *CD* are similar to three parallelograms of the solid *AL*.

But the former three are both equal and similar to the three opposite parallelograms, and the latter three are both equal and similar to the three opposite parallelograms; therefore the whole solid *CD* is similar to the whole solid *AL*. [[book-11/definitions#Definition 9|XI. Def. 9]]

Therefore on the given straight line *AB* there has been described *AL* similar and similarly situated to the given parallelepipedal solid *CD*. Q. E. F.