Maths Teaching Idea #5
A commonly used Circle Theorem in Olympiad problems is known as the Power of a Point Theorem. It isn’t mentioned in the GCSE syllabus and is certainly not referenced in any textbooks that I have used. I feel that it could be more widely explored and investigated in maths classrooms.
The theorem is simple, as per the images above the theorem states that:
- DE . BE = AE . CE
- AB² = BC . DC
- BC . AC = DC . EC
The proof of the theorem requires adding lines to construct similar triangles. This leads to setting up equal ratios and the result follows.
The closest the UK maths curriculum gets to this is a special case of diagram 2/3, which is that two tangents from an external point are equal in length, i.e. CE = CF in the image below. This is an easy application of the theorem.
The Power of a Point theorem strikes me as having an obvious place in the teaching of Circle Geometry and is certainly something that I will be inviting learners to play around with in future. It is familiar enough for it to allow for investigation in the KS4 classroom whilst being alien enough for it to challenge learners.
It can also lead to some really nice problems:
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