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intersecting chords theorem

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Intersecting Chords Theorem

The intersecting chords theorem relates the lengths of the pieces of two non-parallel chords drawn in a circle. The chords are broken at their intersection point, which might be inside the circle or which might require the chords to be extended outside the circle. One of the chords can also be a tangent to the circle.

The intersecting chords theorem is equivalent to Pythagoras' Theorem.

Intersection Inside

IntersectingChordsInside.png

With the chords ABA B and CDC D as above, and the intersection point PP inside the circle, then the intersecting chords theorem states that:

AP×PB=CP×PD A P \times P B = C P \times P D

Intersection Outside

IntersectingChordsOutside.png

With the chords ABA B and CDC D as above, and the intersection point PP outside the circle formed by extending the chords to their intersection, then the intersecting chords theorem again states that:

AP×PB=CP×PD A P \times P B = C P \times P D

Tangential Case

IntersectingChordsTangent.png

With the chord ABA B and a tangent at a point CC as above, and the intersection point PP of the chord and tangent formed by extending the chord to this intersection, then the intersecting chords theorem states that:

AP×PB=CP 2 A P \times P B = C P^2

category: circle theorems

Revised on September 7, 2021 22:29:41 by Andrew Stacey
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