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Although the references to postulates, axioms, and previous propositions are not part of the original text of Euclid's Elements, all the editions I have seen contain them.

In many propositions, it is needed to make adjacent segments, say $PR$ and $PQ$ equal. Examples include:

All the editions linked from this page and more (I do not have enough reputation to post the links to them): John Casey’s English translation, Richard Fitzpatrick’s edition, Johan Ludvig Heiberg's edition, and Józef Czech's Polish translation refer in those cases to Proposition I.3 (“To cut off from the greater of two given unequal straight lines a straight line equal to the less”).

As I understand it, for making such adjacent segments equal, Postulate 3 (“Let it be postulated to draw a circle with any center and radius”) is sufficient. Why is Postulate 3 not referred to in all those editions of Elements instead of Proposition I.3?

[Uncontroversial uses of Proposition I.3 include Proposition I.6, where $DB$ is made equal to $AC$.]

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