Proof: (Euclid)
(related to Proposition: 1.19: Angles and Sides in a Triangle II)
- For if not, $AC$ is certainly either equal to, or less than, $AB$.
- In fact, $AC$ is not equal to $AB$.
- For then angle $ABC$ would also have been equal to $ACB$ [Prop. 1.5].
- But it is not.
- Thus, $AC$ is not equal to $AB$.
- Neither, indeed, is $AC$ less than $AB$.
- For then angle $ABC$ would also have been less than $ACB$ [Prop. 1.18].
- But it is not.
- Thus, $AC$ is not less than $AB$.
- But it was shown that ($AC$) is not equal (to $AB$) either.
- Thus, $AC$ is greater than $AB$.
- Thus, in any triangle, the greater angle is subtended by the greater side.
- (Which is) the very thing it was required to show.
∎
Thank you to the contributors under CC BY-SA 4.0!
![](https://github.com/bookofproofs/bookofproofs.github.io/blob/main/_sources/_assets/images/calendar-black.png?raw=true)
- Github:
-
![bookofproofs](https://github.com/bookofproofs.png?size=32)
- non-Github:
- @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"