Elements I.19

Euclid's Elements: Book I: Proposition 19

Let ABC be a triangle having the angle ABC greater than the angle BCA;

I say that the side AC is also greater than the side AB.

For, if not, AC is either equal to AB or less.

Now AC is not equal to AB; for then the angle ABC would also have been equal to the angle ACB; I. 5 but it is not; therefore AC is not equal to AB.

Neither is AC less than AB, for then the angle ABC would also have been less than the angle ACB; I. 18 but it is not; therefore AC is not less than AB.

And it was proved that it is not equal either. Therefore AC is greater than AB.

Therefore etc.

Q.E.D.