Intersecting Chords

From DT Online


This theorem relates to a characteristic of a cyclic quadlitateral, the diagonals of which are two intersecting chords of the circumscribing circle.

The Intersecting Chords Theorem states that the relationship (ao x ob = co x od) is true for any two intersecting chords, whether or not one of them is a diameter.

Measuring Beam Deflections

The theorem can be useful when measuring the radius of bending of a deflected beam if one of the chords is taken to be the length of the beam e.g.

  • set up a strip of material as a simple beam
  • note the distance between supports (cd)
  • apply a central load to make it bend - or ‘deflect’(Δ) - therefore co = od = ½cd
  • measure the deflection (ob)

Bending Radius (R) can be calculated as follows:

  • ao x ob = co x od
  • (2R-Δ) x Δ = ½cd x ½cd
  • but, for small deflections, Δ2 will be negligible
  • so Bending Radius (R) = cd2 ÷ (8 x Δ)

(see Structures: Or Why Things Don't Fall Down by J.E.Gordon)