And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20] Again, let C measure D; I say that A also measures B. For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D. And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20] And A, E, B are continuously proportional; therefore A also measures B. Therefore etc. Q. E. D.
And, as A is to E, so is C to D; therefore also C measures D. [VII. Def. 20] Again, let C measure D; I say that A also measures B. For, with the same construction, we can in a similar manner prove that A, E, B are continuously proportional in the ratio of C to D. And since, as C is to D, so is A to E, and C measures D, therefore A also measures E. [VII. Def. 20] And A, E, B are continuously proportional; therefore A also measures B. Therefore etc. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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