Proposition: 1.21: Triangles within Triangles

(Proposition 21 from Book 1 of Euclid's “Elements”)

If two internal straight lines are constructed on one of the sides of a triangle, from its ends, the constructed (straight lines) will be less than the two remaining sides of the triangle, but will encompass a greater angle. * For let the two internal straight lines $BD$ and $DC$ have been constructed on one of the sides $BC$ of the triangle $ABC$, from its ends $B$ and $C$ (respectively). * I say that $BD$ and $DC$ are less than the (sum of the) two remaining sides of the triangle $BA$ and $AC$, but encompass an angle $BDC$ greater than $BAC$.


Modern Formulation

In an arbitrary triangle \(\triangle{ABC}\), if two segments are constructed from the vertexes of its base to a point within the triangle, then:

Proofs: 1

Proofs: 1

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



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"