11. COAXIAL CIRCLES

Exploring  

We set up the diagram for this chapter one step at a time.

We have seen that if we take two pairs of points A, B; C, D on a straight line, then there exist points X, Y on the line such that (A, B; X, Y) and (C, D; X Y) are both harmonic. It seems likely that we might find other pairs which ar harmonic conjugates with respect to X, Y ?

We can add circles to this diagram with diameters AB, CD. Drawing circles with diameters determined by the other pairs would now give us a whole family of circles.

Presumably all the circles in this family have the same radical axis?

Now, what about the related set of circles which actually pass through the two limiting points X, Y ? Are they related to the original set in any special way? Look at the angles in which they intersect. What do you think?

These then are some of the issues we explore in this chapter.
                                                                                                                                                                                                                                                                                                            


































Involution

Given points A, B, C, D we know there exists a unique pair of points X, Y : (X, Y; A, B) = –1 = (X, Y; C, D).

More generally now, all pairs of points which separate X, Y harmonically are said to form an involution. Corresponding pairs of points in this involution are called mates.

Given any point P on the line, we can determine its unique harmonic conjugate Q, obtaining the pair of mates (P, Q). In particular, each of X, Y is its own mate; X and Y are called double points in the involution.

[To check this last statement you might look at the cross ratio (X, Y; X, Z). The ratio degenerates horribly in this case, but we need Z = X to prevent the ratio taking the wrong value 0.]

                                                                                                                                                                                


































Coaxial Circles

Let us start with given points X, Y as double points of an involution on a line. Now if (P, Q) is a pair of mates in this involution, draw a circle on PQ as diameter. By choosing different pairs of mates, we thus obtain an infinite set of circles, and the points X, Y are inverse with respect to each circle of the system.

Any two of the circles have X and Y as common inverse points, and so the perpendicular bisector of XY is the radical axis of the two circles. Hence we can talk about the common radical axis of the system, and equally, the common limiting points. For this reason the circles are said to form a coaxial (coaxal) system.




                                                                                                                                                                               




























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Inverse Points and Orthogonality

In the diagram, let P, Q be inverse points with respect to the circle.
Then by definition, OP.OQ = OT 2. If P lies outside the circle, then its inverse Q lies on the polar of P – the line TT'.

Theorem 11:1   Points O, Q are inverse with respect to the circle center P and radius PT.

Proof In the diagram, QPT ~ TPO, so PQ / PT = PT / PO. Hence as required.

Now we need a new diagram ...
Theorem 11:2  Let A, X, Y be collinear. Then X and Y are inverse with respect to circle centre A circle centre B cuts circle centre A orthogonally (at right angles).

Proof    If the circles are orthogonal, then AU is tangent to circle centre B, and so . Hence X, Y are inverse with respect to circle centre A. For the converse, simply reverse the argument.

                                                                                                                                                     
                                                                                          
Circles which cut each other at right angles are said to be orthogonal. In this case, the tangent to each circle at a point of intersection passes through the centre of the other – that is, it lies along the radius.































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Orthogonal Circles

We are given the blue system of coaxial circles with limiting points X, Y. Consider now another family of circles: the yellow circles passing through points X and Y. 

Remembering that X and Y are inverse points with respect to every circle of the blue family, Theorem 11.2 now tells us that every circle in the yellow family cuts every circle in the blue family at right angles. That is, we now have two families of orthogonal circles.

Using our more general interpretation of radical axis to include the common chord of two circles, we can in fact describe this system as being made up of two orthogonal families of coaxial circles. 


                                                                                                                                                     
                                                                                          


























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Extensions

1.  Three pairs of distinct points (P, P'), (Q, Q'), (R, R') are known to be mates in an involution. Show that the three involutions having (P, P'), (Q, Q'), (R, R') respectively as double points, have a pair of mates in common.

2. Show that two involutions on a straight line have precisely one pair of mates in common.

3. Two circles cut in A, B and Q lies on AB produced. How would you construct a circle with Q as centre which cuts each of the given circles orthogonally? Does your construction work if Q lies between A and B?

4. Show that if P is a variable point on a given circle of a coaxial system with limiting points X, Y, then the ratio PX / PY is constant. Rewrite this problem, showing that the question can be expressed much more simply.

5. Given a family of coaxial circles and an arbitrary line, show that two circles of the system touch the line.

                                           Hints and Solutions ...

For looking up ...

http://mathworld.wolfram.com/CoaxalCircles.html

Geometry for advanced pupils, E. A. Maxwell (Oxford 1963)
                                                                                                                                                          
                                                                                          






























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Hints and Solutions

1.  Three pairs of distinct points (P, P'), (Q, Q'), (R, R') are known to be mates in an involution. Show that the three involutions having (P, P'), (Q, Q'), (R, R') respectively as double points, have a pair of mates in common.

• Let A, B be the double points of the given involution. Then (A, B) is the common pair of mates of the final three involutions.

2. Show that two involutions on a straight line have precisely one pair of mates in common.

• Given points A, B, C, D we know there exists a unique pair of points X, Y : (X, Y; A, B) = –1 = (X, Y; C, D).
   So let A, B and C, D be the double points of the given involutions. Then (X, Y) is the unique pair of mates.

3. Two circles cut in A, B and Q lies on AB produced. How would you construct a circle with Q as centre which cuts each of the given circles orthogonally? Does your construction work if Q lies between A and B?

• Construct the polars q, q' of Q with respect to the two circles by completing the quadrangles AB**. These polars meet the circles in points of tangency T, T' from Q. Take QT as the radius of the required orthogonal circle.

4. Show that if P is a variable point on a given circle of a coaxial system with limiting points X, Y, then the ratio PX / PY is constant. Rewrite this problem, showing that the question can be expressed much more simply.

• Let the circle meet XY in A, B. Then we have the circle occurring as an Apollonius circle with X, Y inverse points.

5. Given a family of coaxial circles and an arbitrary line l, show that two circles of the system touch the line.

• Suppose the circles meet in A, B. Each point P on l determines a circle of the system which gives rise toa further point P' on l. Points P, P' are mates in involution, and the double points correspond to circles touching the line.