Proposition: 1.19: Angles and Sides in a Triangle II

(Proposition 19 from Book 1 of Euclid's “Elements” - this a the conversion to the proposition 1.18)

In any triangle, the greater angle is subtended by the greater side. * Let $ABC$ be a triangle having the angle $ABC$ greater than $BCA$. * I say that side $AC$ is also greater than side $AB$.

fig19e

Modern Formulation

In a given triangle \(\triangle{ABC}\) with the angle \(\angle{ABC}\) greater than the angle \(\angle{BCA}\), the side \(\overline{AC}\) opposite to the greater angle is longer than \(\overline{AB}\), opposite to the smaller angle.

Proofs: 1

Proofs: 1 2 3 4 5


Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs
non-Github:
@Calahan
@Casey
@Fitzpatrick


References

Adapted from CC BY-SA 3.0 Sources:

  1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014

Adapted from (Public Domain)

  1. Casey, John: "The First Six Books of the Elements of Euclid"

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"