Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
(Proposition 13 from Book 10 of Euclid's “Elements”)
If two magnitudes are commensurable, and one of them is incommensurable with some magnitude, then the remaining (magnitude) will also be incommensurable with it.
* Let $A$ and $B$ be two commensurable magnitudes, and let one of them, $A$, be incommensurable with some other (magnitude), $C$.
* I say that the remaining (magnitude), $B$, is also incommensurable with $C$.
Modern Formulation
(not yet contributed)
Table of Contents
Proofs: 1
 Lemma: Lem. 10.13: Finding Pythagorean Magnitudes
Mentioned in:
Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016