Activity 1-7: The Overlapping Circles

1. www.carom-maths.co.uk Activity 1-7: The Overlapping Circles

2. Draw three overlapping circles like so in Geogebra:

3. Add the straight lines AB, CD and EF, and then drag... Task: can you PROVE that the three lines always meet?

4. The equation of the green circle AEB we can write as x2 + y2 + a1x + b1y + c1= 0, or C1 = 0. The equation of the blue circle ADB we can write as x2 + y2 + a2x + b2y + c2 = 0, or C2 = 0. What happens if we subtract these two equations? We get (a1 - a2)x + (b1 - b2)y + (c1 - c2) = 0. So we have a straight line – but which line is it?

5. A satisfies both C1 = 0 and C2 = 0, so must be on C1 - C2 = 0. B satisfies both C1 = 0 and C2 = 0, so must be on C1 - C2 = 0. So the line given by C1 - C2 = 0 must be the line AB. By an exactly similar argument, the line given by C2 – C3 = 0 must be the line CD, where C3 = 0 is the equation of the red circle CED, x2 + y2 + a3x + b3y + c3 = 0.

6. Now think about the point where AB and CD meet. This is onC1 – C2 = 0, and also on C2 – C3 = 0. So this point must be on the sum of these equations, which is C1– C3 = 0. But what is this, but the equation of the line EF? Thus we have that the lines AB, CDand EF are concurrent.

7. A new question now: what happens if we ADD the equations of our circles?

8. Clearly C1 + C2 + C3 = 0 is the equation of a circle. Try the Autograph file below that shows what happens when you add three circles like this. Three Circles Autograph File http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-1-7-3.agg Take thegreen, blue and red circles. The purple circle is their sum – the ‘sum-circle’. Task – What do you notice about the sum-circle? can you prove this?

9. The purple circle, the sum-circle, seems to always enclose the intersection of the red, greenand blue circles. Point A is inside the green circle  C1 < 0 for point A. Point B is inside the blue circle  C2 < 0 for point B. Point C is inside the red circle  C3 < 0 for point C. So point D is inside the blue, red and green circles  C1 + C2 + C3 < 0 for point D  point D is inside the sum-circle.

10. Or else you might like to explore Friendly Circles... The file below shows what happens as we vary a, b and c. Friendly Circle Autograph file http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-1-7-4.agg

11. Conjecture: some pair of these three circles always overlap, even if only in a single point. In other words, the three circles are never disjoint. Or we might say, the circles are Friendly. Proof: clearly the theorem is true for (a, b, c) just if it is true for ( a,  b,  c). So we only need to consider the cases where a, b and c are all positive, or where exactly one of a, b andcis negative.

12. Suppose a, b and c are all positive, and suppose the theorem is false. So √((a  b)2 + (b  c)2) > c + a, and squaring gives b2 > ab + bc + ca. Similarly we have a2 > ab + bc + ca, and c2 > ab + bc + ca. Multiplying these gives (abc)2 > (ab + bc + ca)3, which is clearly absurd.

13. Suppose instead that just a is negative, with a = d. Then √(( d  b)2 + (b  c)2) > c + d, which gives b2 + bd  bc > cd. Completing the square we have Now taking the square root, Similarly c > b, which is a contradiction. So the three circles are always friendly.

14. With thanks to: David Sharpe of Mathematical Spectrum for publishing my article on Sumlines, and Shaun Stevens. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net