Proposition: 1.40: Triangles of Equal Area IV

(Proposition 40 from Book 1 of Euclid's “Elements”)1

Equal triangles which are on equal bases, and on the same side, are also between the same parallels. * Let $ABC$ and $CDE$ be equal triangles on the equal bases $BC$ and $CE$ (respectively), and on the same side (of $BE$). * I say that they are also between the same parallels. fig40e

Modern Formulation

Triangles which are equal in area (\(\triangle{ABC}=\triangle{DCE}\)) as well as stand on equal bases (\(\overline{BC}\), \(\overline{CE}\)) and on the same side of their bases stand on the same parallels (\(\overline{AD}\parallel\overline{BE}\)).

Proofs: 1

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. This whole proposition is regarded by Heiberg as a relatively early interpolation to the original text (translator's note).