What are circle theorems?
- Circle theorems are angle (and distance) properties that arise when lines and shapes drawn within or connected to a circle
- There are quite a few of them …
- Circle Theorems – Angles at Centre & Circumference
- Circle Theorems – Tangents
- Circle Theorems – Cyclic Quadrilaterals
- Circle Theorems – Alternate Segment
- This set of notes deals with the Intersecting Chord Theorem
- But first make sure you are familiar with the names of parts of a circle
What is the intersecting chord theorem?
- For two chords, AB and CD that meet at point P
- AP : PD ≡ CP : PB
- Ratio of longer lengths (of chords) ≡ Ratio of shorter lengths (of chords)
- An more practical way to deal with most problems is
- AP ×PB = CP × PD
- You do not need to know the proof this theorem
- You may be able to see a loose connection to similar shapes
How do I use the intersecting chord theorem to solve problems?
- Keep track carefully of which distance is associated with each part of each chord
Harder problems with intersecting chord theorem
- The algebra can be made harder by having more, and more awkward expressions for the distances involved
If you do not like the capital letter notation used you can rename the lengths of the chord using single letters (see the second diagram above).
The multiplication version of the theorem is easier to remember and work with but you may be asked questions about ratios too (see the Worked Example, part (b), below).