---
book: 5
number: 9
id: "V.9"
kind: "theorem"
uses: ["[[book-5/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.9"
license: "CC-BY-SA-4.0"
---

# V.9

*Magnitudes which have the same ratio to the same are equal to one another*; *and magnitudes to which the same has the same ratio are equal*.

## Proof

For let each of the magnitudes *A*, *B* have the same ratio to *C*; I say that *A* is equal to *B*.

For, otherwise, each of the magnitudes *A*, *B* would not have had the same ratio to *C*; [[book-5/proposition-8|V. 8]] but it has; therefore *A* is equal to *B*.

Again, let *C* have the same ratio to each of the magnitudes *A*, *B*; I say that *A* is equal to *B*.

For, otherwise, *C* would not have had the same ratio to each of the magnitudes *A*, *B*; [[book-5/proposition-8|V. 8]] but it has; therefore *A* is equal to *B*.

Therefore etc. Q. E. D.