Proposition: 1.39: Triangles of Equal Area III

(Proposition 39 from Book 1 of Euclid's “Elements”)

Equal triangles which are on the same base, and on the same side, are also between the same parallels.

Modern Formulation

Triangles (\(\triangle{BAC}\), \(\triangle{BDC}\)), which are equal in area and stand on the same base (\(\overline{BC}\)) and on the same side of the base also stand between the same parallels (\(\overline{AD}\), \(\overline{BC}\)).

Proofs: 1

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"