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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. I say that A is also similar to B.
Since A is similar to C, it is equiangular with it and has the sides about the equal angles proportional. | 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 and B is equiangular with C and with C has the sides about the equal angles proportional, therefore A is similar to B. | V.11
Therefore, figures which are similar to the same rectilinear figure are also similar to one another.
Q.E.D.

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This proposition is used in the proofs of propositions VI.24, VI.28, and VI.29.

Book VI Introduction - Proposition VI.20 - Proposition VI.22.

© 1996
D.E.Joyce
Clark University