If three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number.

Let *A, B,* and *C,* three numbers in continued proportion, be the least of those which have the same ratio with them.

I say that the sum of any two of the numbers *A, B,* and *C* is relatively prime to the remaining number, that is, *A* plus *B* is relatively prime to *C, B* plus *C* is relatively prime to *A,* and *A* plus *C* is relatively prime to *B.*

Take two numbers *DE* and *EF* to be the least of those which have the same ratio with *A, B,* and *C.*

Cor. to VIII.2

It is then manifest that *DE* multiplied by itself makes *A,* and multiplied by *EF* makes *B,* and that *EF* multiplied by itself makes *C.*

Now, since *DE* and *EF* are least, therefore they are relatively prime. But, if two numbers are relatively prime, then their sum is also relatively prime to each, therefore *DF* is relatively prime to each of the numbers *DE* and *EF.*

But, further, *DE* is also relatively prime to *EF,* therefore *DF* and *DE* are relatively prime to *EF.* But, if two numbers are relatively prime to any number, then their product is also relatively prime to the other, so that the product of *FD* and *DE* is relatively prime to *EF,* hence the product of *FD* and *DE* is also relatively prime to the square on *EF.*

But the product of *FD* and *DE* is the square on *DE* together with the product of *DE* and *EF,* therefore the sum of the square on *DE* and the product of *DE* and *EF* is relatively prime to the square on *EF.*

And the square on *DE* is *A,* the product of *DE* and *EF* is *B,* and the square on *EF* is *C,* therefore the sum of *A* and *B* is prime to *C.*

Similarly we can prove that the sum of *B* and *C* is relatively prime to *A.*

I say next that the sum of *A* and *C* is also relatively prime to *B.*

Since *DF* is relatively prime to each of the numbers *DE* and *EF,* therefore the square on *DF* is also relatively prime to the product of *DE* and *EF.*

But the sum of the squares on *DE* and *EF* together with twice the product of *DE* and *EF* equals the square on *DF,* therefore the sum of the squares on *DE* and *EF* together with twice the product of *DE* and *EF* is relatively prime to the product of *DE* and *EF.*

Taken separately, the sum of the squares on *DE* and *EF* together with the product of *DE* and *EF* is relatively prime to the product of *DE* and *EF.*

Therefore, taken separately again, the sum of the squares on *DE* and *EF* is relatively prime to the product of *DE* and *EF.*

And the square on *DE* is *A,* the product of *DE* and *EF* is *B,* and the square on *EF* is *C.*

Therefore the sum of *A* and *C* is relatively prime to *B.*

Therefore, *if three numbers in continued proportion are the least of those which have the same ratio with them, then the sum of any two is relatively prime to the remaining number.*

Q.E.D.

where *d* and *e* are relatively prime. Then the sum, *d* + *e,* is relatively prime to both *d* and *e* (VII.28).

Now, since both *d* and *d* + *e* are relatively prime to *e,* so is their product *d*^{2} + *de* relatively prime to *e* (VII.24), and therefore to *e*^{2} (VII.25). Thus, *a* + *b* is relatively prime to *c.*

Likewise, *b* + *c* is relatively prime to *a.*

Next, since *d* + *e* is relatively prime to both *d* and *e,*
so is its square (*d* + *e*)^{2} relatively prime to the product *de* (VII.24 and VII.25). That is, *d*^{2} + *e*^{2} + 2*de* is relatively prime to *de.* Subtract 2*de* to conclude that *d*^{2} + *e*^{2} is relatively prime to *de.* Thus, *b* is relatively prime to *a* + *c.*
Q.E.D.

This proposition is not used in the rest of the *Elements*.