equality of parallelogram complements

../../../../_images/elem.1.prop.43.png

In any parallelogram the complements [^I.43:1] of the parallelograms about the diameter are equal to one another.

===

Let ABCD be a parallelogram, and AC its diameter; and about AC let EH, FG be parallelograms, and BK, KD [^I.43:2] the so-called complements;

I say that the complement BK is equal to the complement KD.

For, since ABCD is a parallelogram, and AC its diameter,

  • the triangle ABC is equal to the triangle ACD. [I.34]

Again, since EH is a parallelogram, and AK is its diameter,

  • the triangle AEK is equal to the triangle AHK.

For the same reason

  • the triangle KFC is also equal to KGC.

Now, since the triangle AEK is equal to the triangle AHK,

  • and KFC to KGC,

the triangle AEK together with KGC is equal to the triangle AHK together with KFC. [I.c.n.2]

And the whole triangle ABC is also equal to the whole ADC; therefore the complement BK which remains is equal to the complement KD which remains. [I.c.n.3]

Therefore etc.

## References

[I.34]: /elem.1.34 “Book 1 - Proposition 34” [I.c.n.2]: /elem.1.c.n.2 “Book 1 - Common Notion 2” [I.c.n.3]: /elem.1.c.n.3 “Book 1 - Common Notion 3”

## Footnotes

[^I.43:1]: complements,

<foreign lang=”greek”>παραπληρώματα</foreign>, the figures put in to fill up (interstices).

[^I.43:2]: and about AC….

Euclid’s phraseology here and in the next proposition implies that the complements as well as the other parallelograms are <quote>about</quote> the diagonal. The words are here <foreign lang=”greek”>περὶ δὲ τὴν ΑΓ παραλληλόγραμμα μὲν ἔστω τὰ ΕΘ, ΖΗ, τὰ δὲ λεγόμενα παραπληρώματα τὰ ΒΚ, ΚΔ</foreign>. The expression <quote>the so-called complements</quote> indicates that this technical use of <foreign lang=”greek”>παραπληρώματα</foreign> was not new, though it might not be universally known.