Proposition 21

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

Let each of the rectilinear figures A and B be similar to C.

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I say that A is also similar to B.

VI.Def.1

Since A is similar to C, it is equiangular with it and has the sides about the equal angles proportional.

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

V.11

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

Therefore, figures which are similar to the same rectilinear figure are also similar to one another.

Q.E.D.

Guide

 This proposition is used in the proofs of propositions VI.24, VI.28, and VI.29. It also would have been useful in the proof of VI.8.