\begin{definition}[Definition:Duplicate Ratio]
Let $a, b, c$ be magnitudes such that:
:$a : b = b : c$
Then $a$ has the '''duplicate ratio''' to $c$ of the ratio it has to $b$.
That is:
:$a : c$ is the '''duplicate ratio''' of $a : b$.
From definition of ratio:
{{begin-eqn}}
{{eqn | l = a : b
      | r = b : c
      | c =
}}
{{eqn | ll= \leadsto
      | l = \dfrac a b
      | r = \dfrac b c
      | c =
}}
{{eqn | ll= \leadsto
      | l = \dfrac a c
      | r = \dfrac a b \dfrac b c
      | c =
}}
{{eqn | ll= \leadsto
      | l = \dfrac a c
      | r = \paren {\dfrac a b}^2
      | c =
}}
{{end-eqn}}
That is:
: $a : c = \paren {a : b}^2$
{{EuclidSaid}}
:''{{Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio}}
{{EuclidDefRefNocat|V|9|Duplicate Ratio}}
Category:Definitions/Euclidean Algebra
\end{definition}