From symmetry to circle theorems

As I’m reading through different research papers, pedagogical guides and textbooks, I can often plot a mathematical path through the school years. This is one of the beautiful things I have an opportunity to do – join together what are seemingly disjoint ideas in primary school with challenging geometrical problems pupils might not meet until secondary. 

One such case is that of symmetry, constructions and circle theorems. Below I’ve pulled together work by Fujita and Jones (2002), Godfrey and Siddons (1912), and Mason (2010). I should also at this point recognise the work of Dr Nicola Bretscher, UCL Institute of Education – participating in her workshops on circle theorems using Geogebra has been incredibly helpful in understanding how carefully designed informal and formal construction tasks can reveal beautiful results. 

Let’s assume you have not yet met any circle theorems or formal constructions, but have mentally established the concept of a circle as the set of points equidistant from a centre. Here’s a series of tasks to consider. They are worth completing as you go – no cheating and reading ahead! These tasks are designed to cover a series of years: with each visit refining ideas working towards formalised procedures with understanding. For me they highlight the need for making connections between areas of mathematics and making sure pupils can explain what’s going on. 

Let’s start by investigating mirror lines. What geometric relationships exist between the mirror line and pairs of corresponding points on an object and its image? Pick any point on the mirror line and join it with line segments to a point on the object and the corresponding point on its image. What do you notice about the line segments? So how can you use circles to construct points on the mirror line? How many points would you need to identify to find the mirror line? Why? 

Following this train of thought, you’ve established that any point on a mirror line is exactly the same distance between any pair of corresponding points on the object and its image. A nice way to think about – or experience – this is by folding along the mirror line: corresponding points overlap with each other, so joining them to any point on the mirror line makes it clear that they are the same distance away from that point. 

One way of defining a line in a particular position is by finding (a minimum) of two points that lie on the line. Hence, we can construct the mirror line if we can find two points on it. This can be achieved by constructing two circles, each having the same radius, one with a point on the object as its centre and the other with the corresponding image point as its centre – around two corresponding points on the object and its image – so long as the circles intersect twice. An alternative is to construct two pairs of matching arcs – they don’t even have to be from the same corresponding points. 

So we’ve constructed the mirror line. If we pick any pair of corresponding points they are both the same distance to any point on this line. 

When would these circles only touch once? Why? What does this tell you about a radius and tangent (a line that touches the circumference but does not cross it)? Use the diagram below to help explain what’s happening: 

The shortest distance between two points is the straight line between them (the dotted line in the image below) so the circles touch once when their radius is half of this distance (remember the circles are identical) – any shorter and the circles won’t meet. Now let’s think of another point on the mirror line and join it to the corresponding points on the object and its image. What can you say now? Use the discussions above (and your knowledge of isosceles triangles) to justify why the dotted line and mirror line are perpendicular to each other. Hence you’ve constructed the perpendicular bisector between the two points, and found that a tangent meets a radius at 90˚. 

Embedded in this train of thought is also information about a chord and the radius. Let’s start with one circle touching the mirror line and increase its radius. 

Analysing this diagram is fascinating. The chord (now in red) is perpendicular to the radius, its ends are the same distance from the indicated point on the object and the image, so actually the dotted line is now a mirror line between the two ends of the chord – hence it’s a perpendicular bisector of the chord. In other words the perpendicular bisector of a chord passes through the centre of the circle. 

Actually investigating and pulling this construction apart is extremely powerful. Dig down even further and you’ve got the construction of a rhombus via its diagonals (recalling that the diagonals of a rhombus are perpendicular bisectors of one another). 

Opening up pupils’ minds to these kinds of experiences starts in the earliest years. We ask pupils what they notice about reflections. What happens if a shape moves closer to the mirror? They become accustomed to noticing and encouraged to explain and justify whether they are convinced. High quality examples to discuss and teachers who can see the whole picture of development are essential – and, where possible, resources such as dynamic geometry environments further aid the progression.