For, if they are not prime to one another, some number will measure them.
Let a number measure them, and let it be D.
Now A is odd;
therefore D is also odd.
And since D which is odd measures C, and C is even,
therefore [D] will measure the half of C also. [IX. 30]
But B is half of C;
therefore D measures B.
But it also measures A;
therefore D measures A, B which are prime to one another: which is impossible.
Therefore A cannot but be prime to C.
Therefore A, C are prime to one another.
Q. E. D.