# 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: $\angle{BDC} > \angle{BAC}.$

Proofs: 1

Proofs: 1

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### References

#### Adapted from CC BY-SA 3.0 Sources:

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