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

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

If not, either line AC equals line AB or it is less than it.

Now line AC does not equal line AB, for then the angle ABC would equal the angle ACB, but it does not. Therefore line AC does not equal line AB.

Neither is line AC less than line AB, for then the angle ABC would be less than the angle ACB, but it is not. Therefore line AC is not less than line AB.

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

Therefore in any triangle the side opposite the greater angle is greater.

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