---
book: 6
number: 21
id: "VI.21"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:6.prop.21"
license: "CC-BY-SA-4.0"
---

# VI.21

*Figures which are similar to the same rectilineal figure are also similar to one another*.

## Proof

For let each of the rectilineal figures *A*, *B* be similar to *C*; I say that *A* is also similar to *B*.

For, since *A* is similar to *C*, it is equiangular with it and has the sides about the equal angles proportional. [[book-6/definitions#Definition 1|VI. Def. 1]]

Again, since *B* is similar to *C*, it is equiangular with it and has the sides about the equal angles proportional.

Therefore each of the figures *A*, *B* is equiangular with *C* and with *C* has the sides about the equal angles proportional; therefore *A* is similar to *B*. Q. E. D.