Proposition: 1.22: Construction of Triangles From Arbitrary Segments

(Proposition 22 from Book 1 of Euclid's “Elements”)

To construct a triangle from three straight lines which are equal to three given [straight lines]. It is necessary for (the sum of) two (of the straight lines) taken together in any (possible way) to be greater than the remaining (one), [on account of the (fact that) in any triangle (the sum of) two sides taken together in any (possible way) is greater than the remaining (one) [Prop. 1.20] ].


Modern Formulation

If for three arbitrary segments $A=\overline{DF}$, $B=\overline{FG}$, and $C=\overline{GH}$, the sum of every two pairs of segments is greater than the length of the remaining segment, then it is possible1 to construct a triangle, whose three sides are respectively equal to the three segments.

Proofs: 1

Proofs: 1
Sections: 2

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"


  1. Please note that this is the conversion of the triangle inequality