Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram

(Proposition 33 from Book 1 of Euclid's “Elements”)

Straight-lines joining equal and parallel (straight lines) on the same sides are themselves also equal and parallel. * Let $AB$ and $CD$ be equal and parallel (straight lines), and let the straight lines $AC$ and $BD$ join them on the same sides. * I say that $AC$ and $BD$ are also equal and parallel.


Modern Formulation

Let \(\overline{AB}\), \(\overline{CD}\) be two equal parallel segments. Then the segments joining their adjacent endpoints, (\(\overline{AC}, \overline{BD}\)) are themselves parallel and equal in length: \(\overline{AC}=\overline{BD}\) and \(\overline{AC}\parallel\overline{BD}\). In other words, the quadrilateral \({ABCD}\) forms a parallelogram.

Proofs: 1

Proofs: 1 2 3 4 5 6

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"