Proposition 19

Mon, 2006-10-23 15:21 — ashley

In any triangle the greater angle is subtended by the greater side.

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 $ACB$ ; [Prop. 1.5]

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 $ACB$ ; [Prop 1.18]

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.

AshleyMills.com