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.
For let each of the magnitudes A, B have the same ratio to C;
I say that A is equal to B.
For, otherwise, each of the magnitudes A, B would not have had the same
ratio to C;
but it has; therefore A is equal to B.
Again, let C have the same ratio to each of the magnitudes A, B;
I say that A is equal to B.
For, otherwise, C would not have had the same ratio to each of the magnitudes
A, B;
but it has; therefore A is equal to B.
Therefore etc.
Q.E.D