Proposition: 1.41: Parallelograms and Triagles

(Proposition 41 from Book 1 of Euclid's “Elements”)

If a parallelogram has the same base as a triangle, and is between the same parallels, then the parallelogram is double (the area) of the triangle.


Modern Formulation

If a parallelogram (\(\boxdot{ABCD}\)) and a triangle (\(\triangle{EBC}\)) stand on the same base (\(\overline{BC}\)) and between the same parallels (\(\overline{AE}\), \(\overline{BC}\)), then the parallelogram is double the area of the triangle.

Proofs: 1

Proofs: 1 2 3

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"