Proposition: 3.35: Intersecting Chord Theorem

(Proposition 35 from Book 3 of Euclid's “Elements”)

If two straight lines in a circle cut one another then the rectangle contained by the pieces of one is equal to the rectangle contained by the pieces of the other.


Modern Formulation

If in a circle two chords $\overline{AC},$ $\overline{BD}$ intersect at the point $E$ and let the lengths of segments be defined by $|\overline{AE}|$, $|\overline{EC}|$, $|\overline{DE}|$, and $|\overline{EB}|.$ Then the respective rectangles built from these segments have equal areas, i.e. $$|\overline{AE}|\cdot|\overline{EC}|=|\overline{DE}\cdot|\overline{EB}|.$$

Proofs: 1

Sections: 1

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



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016