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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

Thank you to the contributors under CC BY-SA 4.0!

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non-Github:

@Calahan

@Casey

@Fitzpatrick

References

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"