The Circle of Apollonius - Another way of looking at Circles

 

We learn at school that...

A circle is a plane figure bounded by one curved line such that all straight lines, drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.      — Euclid, Book 1, Elements.

Circles have no edges, vertices or corners - they are perfectly smooth.  Among all closed shapes with the same perimeter, the circle encloses the greatest possible area (the isoperimetric theorem). Circular shapes frequently appear in nature, art, engineering, architecture etc. 

A circle has no beginning and no end- it symbolizes perfection and eternity. (see also)

This is how I had always viewed a circle  -  a two dimensional round shape where every point on its perimeter is the same distance from the centre.  That is until, by chance, I met Apollonius in my local library. I have been walking the corridors of science departments of universities all over the world for the past 66 years and still did not know about Apollonian circles. I wish to set this right.

Slide 0:

The Circle of Apollonius:  is the set of all points (P) where the distance (PB) to one fixed point (B) divided by the distance (PA) to a second point (A) equals a constant positive number k.

For k ≠ 1, the set of points (locus of P) forms a true circle surrounding the point with the shorter of the lengths PA and PB.  The centre of the circle lies on the line passing through A and B.  

For k= 1, the set of points (locus of P) forms a straight line that is the perpendicular bisector of the line segment between A and B.

First, I would discuss some important characteristics of  Apollonian circles.  Then we shall look at the analysis to show that the locus of points P is indeed a circle.

Slide 1:  

Points to note:

• Points A and B are the foci of the circle, and lengths PA and PB are called the focal radii. 
• The line connecting the two foci intersects the circle at two points (C and D) with CD equal to the diameter of the circle (= 2R). 
• The focal radii AP and BP form a right angle at P (∡ APB = 90 degrees).  (more on this later)
• For k = 1, AP = BP.  The locus of P is a straight line that is the perpendicular bisector of the line AB. This is demonstrated in the slides below.

  Slide 2:

Slide 3:

Family of Apolonian Circles:  Apollonius actually defined a second set of circles that pass through the two points A and B. This set of circles cross the first set (described above) orthogonally, viz., the tangent lines drawn to each circle at the intersection point are perpendicular to each other.  This is shown in the next slide.

Slide 4:  

A Proof of Apollonian Circles:  In the following, I show that points P as described above (slide 1) indeed describe a circle.  There are some elegant proofs available in the literature, but I feel these require a good background in mathematics.  Here, I shall describe a simpler method - surprisingly I have not come across this method in publications accessible to me.

Given a line segment between two points A and B, it is always possible to construct a circle that intersects the line passing through A and B at two points C and D.  Points C and D are such that BC and BD are k times AC and AD respectively.  This is explained in the next three slides.

 Slide 5: 

Slide 6:

Slide 7:

Slide 8:

The algebra for k <1 follow exactly the same steps but point B is swapped with point A.  The Apollonian circle encloses point B for k <1.  Also see Slide 2.
 

A Geometrical Proof of the Appolonian Circles:

In the following, I discuss quite a neat proof that is based on geometry - in particular on the angle bisector theorem.  This theorem is not well known and  I shall first discuss it in slides 9,10 & 11.  It is possible to jump to slide 12 without loss of continuity - though, the angle bisector theorem gives a good taste of how geometry works.  

Slide 9:

Slide 10: 

Slide 11:

Having established the angle bisector theorem, we shall use it to define the Apollonian circles.

Slide 12:

Slide 13:

Slide 14:

Appendix

End Note:  The mathematical prowess of the ancient Greek mathematicians was amazing - particularly in Geometry.  I had never imagined that the simplest shape of a circle can be so exciting to analyse.  I found working on the Apollonian circles feature so satisfying - not only because it provides an alternate way of defining a circle but also the analysis around  the topic only involves the use of simple algebra and geometry - just goes to show that the maths we learn at the high school level is already very powerful and may be used to provide deep insights in to how things work around us. 

Apollonian circles are very useful and for those who wish to learn more about them, I shall refer to some nice descriptions available (1, 2, 3).