◀ ▲ ▶Branches / Geometry / Elements-euclid / Book--6-similar-figures / Proposition: 6.16: Rectangles Contained by Proportional Straight Lines

Proposition: 6.16: Rectangles Contained by Proportional Straight Lines

(Proposition 16 from Book 6 of Euclid's “Elements”)

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

• Let $AB$, $CD$, $E$, and $F$ be four proportional straight lines, (such that) as $AB$ (is) to $CD$, so $E$ (is) to $F$
• I say that the rectangle contained by $AB$ and $F$ is equal to the rectangle contained by $CD$ and $E$ (and conversely).

Modern Formulation

With $a:=|\overline{AB}|,$ $b:=|F|,$ $c:={|\overline{CD}|}$, and $d:=|E|,$ this proposition states that $\frac ac=\frac db$ if and only if $ab=cd.$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5 6 7

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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