Proposition: 5.09: Magnitudes with Same Ratios are Equal

(Proposition 9 from Book 5 of Euclid's “Elements”)

(Magnitudes) having the same ratio to the same (magnitude) are equal to one another. And those (magnitudes) to which the same (magnitude) has the same ratio are equal. * For let $A$ and $B$ each have the same ratio to $C$. * I say that $A$ is equal to $B$. * So, again, let $C$ have the same ratio to each of $A$ and $B$. * I say that $A$ is equal to $B$.

Modern Formulation

In modern notation, this proposition reads that if $$\alpha =\beta$$ then $\frac\alpha\gamma = \frac\beta\gamma$ and if $\frac\gamma\alpha = \frac\gamma\beta$ then $$\alpha =\beta$$ for all positive real numbers $$\alpha,\beta,\gamma$$.

Generalized Formulation

The above proposition is true for all real numbers with $$\alpha\neq 0, \beta\neq 0, \gamma\neq 0$$, since it follows from the existence and uniqueness of inverse real numbers with respect to multiplication.

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Sections: 17

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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