Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
(Proposition 33 from Book 1 of Euclid's “Elements”)
Straightlines joining equal and parallel (straight lines) on the same sides are themselves 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.
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3 4 5 6
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References
Adapted from CC BYSA 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"