Magnitudes with Same Ratios are Equal

Theorem

As Euclid defined it:

Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.

(The Elements: Book V: Proposition $9$)

That is:

$a : c = b : c \implies a = b$

$c : a = c : b \implies a = b$

Proof

Let each of $A$ and $B$ have the same ratio to $C$, i.e. $A : C = B : C$.

Suppose $A \ne B$.

Then from Relative Sizes of Ratios on Unequal Magnitudes $A : C \ne B : C$.

But $A : C = B : C$ so therefore it is not the case that $A \ne B$.

Therefore $A = B$.

Again, let $C$ have the same ratio to each of $A$ and $B$, i.e. $C : A = C : B$.

Suppose $A \ne B$.

Then from Relative Sizes of Ratios on Unequal Magnitudes $C : A \ne C : B$.

But $C : A = C : B$ so therefore it is not the case that $A \ne B$.

Therefore $A = B$.

$\blacksquare$

Historical Note

This is Proposition 9 of Book V of Euclid's The Elements.

This is the converse of Book V Proposition $7$: Ratios of Equal Magnitudes.