Parts have the same ratio as their equimultiples.

Let *AB* be the same multiple of *C* that *DE* is of *F.*

I say that *C* is to *F* as *AB* is to *DE.*

Since *AB* is the same multiple of *C* that *DE* is of *F,* as many magnitudes as there are in *AB* equal to *C,* there are also in *DE* equal to *F.*

Divide *AB* into the magnitudes *AG, GH,* and *HB* equal to *C,* and divide *DE* into the magnitudes *DK, KL,* and *LE* equal to *F.* Then the number of the magnitudes *AG, GH,* and *HB* equals the number of the magnitudes *DK, KL,* and *LE.*

And, since *AG, GH,* and *HB* equal one another, and *DK, KL,* and *LE* also equal one another, therefore *AG* is to *DK* as *GH* is to *KL,* and as *HB* is to *LE.*

Therefore one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. Therefore *AG* is to *DK* as *AB* is to *DE.*

But *AG* equals *C* and *DK* equals *F,* therefore *C* is to *F* as *AB* is to *DE.*

Therefore, *parts have the same ratio as their equimultiples.*

Q.E.D.

This proposition is used in the next one and a few others in Books V, VI, and XIII.