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:
- The sum of these inner sides will be less than the sum of the outer sides excluding the base: $\overline{AB}+\overline{AC} > \overline{DB}+\overline{DC}.$
- These inner sides will contain a greater angle than the corresponding sides of the outer triangle: $\angle{BDC} > \angle{BAC}.$
Table of Contents
Proofs: 1
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Proofs: 1
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References
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
