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.
* Let $ABCD$ be the given rectilinear figure, and $E$ the given rectilinear angle.
* So it is required to construct a parallelogram equal to the rectilinear figure $ABCD$ in the given angle $E$.
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 $\angle{E}.$
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2 3
Sections: 4
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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"