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In my last blog on this theme I mapped out a journey through a range of discussions and activities looking towards the development of some of the circle theorems starting with considering the mirror line. In this blog I describe a similar journey again developed from the work of Fujita and Jones (2002), Godfrey and Siddons (1912), and Mason (2010).
It’s a practical, hands-on approach where pupils develop an understanding of, and explanation for, the construction of an angle bisector and several circle theorems. Again the concept of a circle as the set of points equidistant from the centre is essential and I would recommend working through the activities yourself to gain a deep understanding of how the story develops.
Start by sketching this diagram freehand. Each circle touches the two line segments that meet at a point.
What is the locus of the centres of the circles? If you folded along this locus what would happen? Test out your ideas.
Why does this happen?
Consider the diagram below and compare lengths OA and OB, AC and BC and consequences of these facts.
This is a lovely activity, because there’s so much going on here…
Folding along the blue dotted line is folding along a diameter, hence a mirror line of the circle itself. AC and BC are radii of the same circle so equal in length; folding along the locus of the centres of the circles A and B shows they coincide, so OA and OB are equal in length and angles AOC and COB are equal in size. Now thinking about consequences of these; triangles OAC and OBC are congruent, you can ‘see’ this when you fold along the line but also because all three sides match up. In my previous blog we established that a tangent meets a radius at right angles, hence angles OAC and OBC are both 90˚
How can we use these facts to construct the angle bisector?
What circle theorems have we uncovered?
I’ll leave you to think about these – please do leave a comment below about what your next step would be.
This story isn’t to be completed in a day, but over several visits highlighted in other lessons – with each visit it’s refined and gradually formalised. By turning some of these problems around and cleverly choosing/designing a starting place – often an informal construction – we open up a whole wonderful world of circle theorems and constructions. As I develop the geometry area of the Cambridge Mathematics Framework I’m really beginning fully to understand the importance of constructions: how they allow a much deeper understanding of geometry and how integral they are to the subject. They are not something simply to copy and repeat procedurally without understanding, but instead offer a window into how shapes are built and their constituent parts. As I delve into higher-level mathematics the ideas established lower down are developed, refined and abstracted, based on these solid foundations.
SOMETHING TO TRY
KS1: Using either pairs of compasses or string and drawing pins, ask pupils to recreate the following diagrams. Discuss what is important about the way in which the circles are constructed; e.g. where their centres are, how their radii compare.
KS2: Using pairs of compasses, ask pupils to recreate the following diagrams exactly (you’ll need to print them out large enough to recreate). Discuss what is important about the way in which the circles are constructed e.g. where their centres are, how their radii compare, what measurements were taken (these are pretty tricky!)
KS3: Analyse the diagram below. Write down as many facts as you can about the diagram (with your reasons) and questions you have.
KS4: An excircle of a triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. For a triangle construct the three excircles.
KS5: This video shows how to construct a regular pentagon:
Explain how and why it works.