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Computable Euclid

Proposition 43

Theorem

Two lines (EF, GH) each parallel to one pair of parallel sides of a parallelogram (◇ABCD), and passing through a point (K) on a diagonal (AC) of the parallelogram, divide it into four parallelograms, of which the two (◇EBGK, ◇HKFD) through which the diagonal does not pass, and which are called the complements of the other two, have the same area.

Commentary

  • Let ◇ABCD be a given parallelogram where AC is a diagonal of ◇ABCD and let K be a point on AC. Through K, construct EF parallel to AD and GH parallel to AB.
  • Then, ◇ABCD is divided into four parallelograms: ◇AEKH, ◇EBGK, ◇HKFD and ◇KGCF. The parallelograms ◇EBGK and ◇HKFD that the diagonal AC does not pass through are called the complements of the other two parallelograms (◇AEKH and ◇KGCF).
  • Then ◇EBGK and ◇HKFD are equal in area.
  • Euclid used the word "diameter" to denote a line joining opposite vertices of a parallelogram; the modern term is "diagonal."
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    Original statement

    παντὸς παραλληλογράμμου τῶν πϵρὶ τὴν διάμϵτρον παραλληλογράμμων τὰ παραπληρώματα ἴσα ἀλλήλοις ἐστίν.

    English translation

    In any parallelogram the complements of the parallelograms about the diameter are equal to one another.

    Computable version

    Additional instances

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