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Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment

(Proposition 5 from Book 13 of Euclid's “Elements”)

If a straight line is cut in extreme and mean ratio, and a (straight line) equal to the greater piece is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater piece.

• For let the straight line $AB$ have been cut in extreme and mean ratio at point $C$.
• And let $AC$ be the greater piece.
• And let $AD$ be [made] equal to $AC$.
• I say that the straight line $DB$ has been cut in extreme and mean ratio at $A$, and that the original straight line $AB$ is the greater piece.

Modern Formulation

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016