Medial or Nine Point Circle

Geometry

CME, MMU, Institute of Education, Didsbury

 

Families of Tangent Circles, the Apollonius Problem: Gergonne's Resolution

The film attempts to reveal a range of properties about circles touching 2 and 3 circles.

The first section reveals some basic facts about the family of circles (coloured red) tangent to two given circles (blue). In order to keep the animation brief only one of the cases is shown (a circle touches 2 others in the “same way” and the role of the direct centre of similarity) although the same properties apply to unlike points of contact and the indirect centre. This insight is needed to fully discuss the 3 circle case. My choice has been to restrict the film to, what I imagine could be called, the paradigmatic case entirely, even though to think this through one needs some insights from the other case.

Section 1

In order these are:

As a member of the tangential family moves continuously through the possibilities the secants through the points of contact of the members of the family of circles are shown to pass through a fixed point which is one of the centres of similarity. For the case shown this is revealed by drawing a common tangent to the blue circles as a red circle passes through this possibility.

A pair of circles from the red family is then shown. One, arbitrary, member is left fixed as another moves. The radical axis of this pair is shown turning whilst always passing through the same similarity centre.

The locus of the intersections of the tangents to the circles at the points of contact is revealed as a straight line. This is the radical axis of the blue pair. Or, the pole of the secant joining the points of contact lies on the radical axis.

These are, almost, all of the facts needed to explain how Gergonne’s method resolves the Apollonius construction problem, though they also need to be perceived as being true for circles touching 2 circles in “different ways” and the The facts are easily proved using standard ratio approaches and so on and are left to be so explained.

The next section shifts to show the 3 circle configuration.

Section 2

A third blue circle appears and one of the members of the first family moves to simultaneously touch all three circles and remains fixed in this position. The radical axis of these red circles is also still shown. As the other circle moves the radical axis is shown as before turning about a similarity centre. When the moving circle also touches all 3 blue circles it changes to being a member of the family touching a different pair of blue circles. The radical axis now turns about a different centre. This is repeated for the remaining pair of blue circles. At the end the 2 circles which are tangential to the 3 given circles and their radical axis are shown. The axis passes through all 3, collinear, similarity centres. The other 2 radical axes of the blue circles taken in pairs appear, concurrent at the radical centre.

Repeating this for the other cases of circles touching 2 circles would show the other 3 collinearities of the similarity centres.

The final section reveals some properties of the configuration of 2, red, circles touching in the “same way’ 3 given, blue, circles.

Section 3

This section attempts to force a shift of attention from perceiving the red circles as tangent to 2 and 3 blue circles to regarding the blue circles as 3 particular members of the family of circles tangent to 2 red circles. The radical centre of the blue circles is shown throughout.

One of the blue circles moves tangent to the 2 red circles, leaving a paler blue trace behind. Tangents to this circle, of course, intersect on the radical axis of the red circles, the line of the 3 similarity centres. The secant of the points of contact with the red circles passes through one of the centres of similarity of the red circles which is shown to be the radical centre. To explain the fact that the radical centre of the 3 blue circles is a similarity centre of the red tangent circles I need to consider inverting the configuration with respect to a circle centred at the radical centre and orthogonal to the 3 blue circles. The blue circles are invariant and the red circles are interchanged. A well known fact about inversion is that the centre of a circle of is a similarity centre of a pair of circles inverted with respect to that circle. I do have an animation showing this, but it adds to the length of the film and has an overly didactic feel to it as a part of this sequence.

Once the moving blue circle has returned to its initial position tangents to this circle are drawn from a point on the radical axis. As this moves along the axis the secant through the points of contact is shown passing through the pole of the radical axis. The pole of the secants lies on the radical axis, and the pole of the radical axis lies on the secant of the tangential points.

Added Notes

A couple of further sections could be added to the film. I am unsure whether they would just make the film longer without adding much or whether they might be useful.

2a

Would show the 3 other possibilities for tangent circles ending with the full set of 4 collinearities. For brevity the configurations could be revealed statically, alluding to the possibility of seeing them in the same way. I have not done this, although these are contained in the short animation corresponding to 3b (below).

3a

Would invert the system with respect to a circle centred at the radical centre animated as the circle changes size. But see my comments after 4.

3b

Show the Gergonne resolution of the problem. Show the 4 lines of the similarity centres and the radical axis. Show, successively, the poles of these lines and the resulting tangent circles in pairs and the full set of 4 families of pairs of circles tangent to 3 circles.

4

I also produced an animation showing the blue circles moving into different configurations together with the full set of tangent circles. It was pretty horrid! Whilst it is easy to produce there seems to me to be no point to it. It is much better to work on a DGS produced figure.

Usefully, it helped me to harden up my position on the differences between animations and draggable DGS figures. I think that the classic films still have an immense value which is not available from a DGS. Films place the viewer in a position of having to sit through an authored sequence without being able to alter it. For this to be valuable the film needs to offer something. A sequence of changing images can offer a variation of perspectives and surprises – I think mine does have some value! The difficulty with an animation such as 4 is that there were no surprises which a user would not naturally create and perceive for themselves. It seems condescending, now, that I even thought of making it. Such a film might have had value before we had Cabri and the other DGSs.