Proposition: 1.45: Construction of Parallelograms III

(Proposition 45 from Book 1 of Euclid's “Elements”)

To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.


Modern Formulation

Given an arbitrary convex quadrilateral $ABCD$ and an arbitrary angle (\(\angle{E}\)), it is possible to construct a parallelogram $FKML$ equal in area to the given \(n\)-sided figure $ABCD,$ which contains an angle $\angle{FKM}$ equal to the given angle $\angle{E}.$

Proofs: 1

Proofs: 1 2 3
Sections: 4

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"