To describe a rectilinear figure similar and similarly situated to a given rectilinear figure on a given straight line.

Let *AB* be the given straight line and *CE* the given rectilinear figure.

It is required to describe on the straight line *AB* a rectilinear figure similar and similarly situated to the rectilinear figure *CE.*

Join *DF.* Construct the angle *GAB* equal to the angle at *C,* and the angle *ABG* equal to the angle *CDF,* on the straight line *AB* at the points *A* and *B* on it.

Therefore the remaining angle *CFD* equals the angle *AGB.* Therefore the triangle *FCD* is equiangular with the triangle *GAB.*

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

Again, construct the angle *BGH* equal to the angle *DFE,* and the angle *GBH* equal to the angle *FDE,* on the straight line *BG* and at the points *B* and *G* on it.

Therefore the remaining angle at *E* equals the remaining angle at *H.* Therefore the triangle *FDE* is equiangular with the triangle *GBH.* Therefore, proportionally, *FD* is to *GB* as *FE* is to *GH,* and as *ED* is to *HB.*

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

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

For the same reason the angle *CDE* also equals the angle *ABH.*

And the angle at *C* also equals the angle at *A,* and the angle at *E* equals the angle at *H.*

Therefore *AH* is equiangular with *CE,* and they have the sides about their equal angles proportional. Therefore the rectilinear figure *AH* is similar to the rectilinear figure *CE.*

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

Q.E.F.

Although the figure has only four sides, it is clear that the method applies to figures with more than four sides.

This proposition is used in the proofs of propositions VI.22, VI.25, and VI.28, and the corollary is used in XII.17.