◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--6-similar-figures / Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines

Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines

(Proposition 17 from Book 6 of Euclid's “Elements”)

If three straight lines are proportional then the rectangle contained by the (two) outermost is equal to the square on the middle (one). And if the rectangle contained by the (two) outermost is equal to the square on the middle (one) then the three straight lines will be proportional.

• Let $A$, $B$ and $C$ be three proportional straight lines, (such that) as $A$ (is) to $B$, so $B$ (is) to $C$.
• I say that the rectangle contained by $A$ and $C$ is equal to the square on $B$ (and conversely).

Modern Formulation

With $a:=|A|$, $b:=|B|$, $c:=|C|$, this proposition states that $\frac ab=\frac bc$ if and only if $ac=b^2.$

Table of Contents

Proofs: 1

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

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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