---
book: 6
number: 18
id: "VI.18"
kind: "construction"
uses: ["[[book-1/proposition-23]]", "[[book-1/proposition-32]]", "[[book-6/proposition-4]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.18"
license: "CC-BY-SA-4.0"
---

# VI.18

*On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure*.

## Proof

Let *AB* be the given straight line and *CE* the given rectilineal figure; thus it is required to describe on the straight line *AB* a rectilineal figure similar and similarly situated to the rectilineal figure *CE*.

Let *DF* be joined, and on the straight line *AB*, and at the points *A*, *B* on it, let the angle *GAB* be constructed equal to the angle at *C*, and the angle *ABG* equal to the angle *CDF*. [[book-1/proposition-23|I. 23]]

Therefore the remaining angle *CFD* is equal to the angle *AGB*; [[book-1/proposition-32|I. 32]] therefore the triangle *FCD* is equiangular with the triangle *GAB*.

Therefore, proportionally, as *FD* is to *GB*, so is *FC* to *GA*, and *CD* to *AB*.

Again, on the straight line *BG*, and at the points *B*, *G* on it, let the angle *BGH* be constructed equal to the angle *DFE*, and the angle *GBH* equal to the angle *FDE*. [[book-1/proposition-23|I. 23]]

Therefore the remaining angle at *E* is equal to the remaining angle at *H*; [[book-1/proposition-32|I. 32]] therefore the triangle *FDE* is equiangular with the triangle *GBH*; therefore, proportionally, as *FD* is to *GB*, so is *FE* to *GH*, and *ED* to *HB*. [[book-6/proposition-4|VI. 4]]

But it was also proved that, as *FD* is to *GB*, so is *FC* to *GA*, and *CD* to *AB*; therefore also, as *FC* is to *AG*, so is *CD* to *AB*, and *FE* to *GH*, and further *ED* to *HB*.

And, since the angle *CFD* is equal to the angle *AGB*, and the angle *DFE* to the angle *BGH*, therefore the whole angle *CFE* is equal to the whole angle *AGH*.

For the same reason the angle *CDE* is also equal to the angle *ABH*.

And the angle at *C* is also equal to the angle at *A*, and the angle at *E* to the angle at *H*.

Therefore *AH* is equiangular with *CE*; and they have the sides about their equal angles proportional; therefore the rectilineal figure *AH* is similar to the rectilineal figure *CE*. [[book-6/definitions#Definition 1|VI. Def. 1]]

Therefore on the given straight line *AB* the rectilineal figure *AH* has been described similar and similarly situated to the given rectilineal figure *CE*. Q. E. F.