Proposition: 1.43: Complementary Segments of Parallelograms

Euclid's Formulation

For any parallelogram, the complements of the parallelograms about the diagonal are equal to one another.


Modern Formulation

Let \(\boxdot{ABCD}\) be a parallelogram with a diagonal \(\overline{AC}\) and let \(K\) be any point on that diagonal. Then segments, which are parallel to the sides of the parallelogram and pass through \(K\), (specifically \(\overline{EF}\), \(\overline{GH}\)) divide \(\boxdot{ABCD}\) into four smaller parallelograms: the two segments through which the diagonal does not pass (\(\boxdot{EBGK}\), \(\boxdot{HKFD}\)) are called the complements of the other two (\(\boxdot{AEKH}\), \(\boxdot{KGCF}\)). Moreover, the complements are equal in area: \(\boxdot{EBGK}=\boxdot{HKFD}\).

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

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"