ON THE GEOMETRY OF MUTUALLY TANGENT CIRCLES AND CYCLOTOMY

ON THE GEOMETRY OF MUTUALLY TANGENT CIRCLES AND CYCLOTOMY

Education for Specialized Subject matter and field Natural Science Education (Mathematics)

Supen'isor:

Matsuoka Takashi Kamwana Evans Muzala

In order to properly study the radii and curvatures of mutually tangent

circles, establish theorems and prove them, we begin with the proof of the simple relationship between the area fl of any triangle ABC to its inradius

and semi-perimeter, that

~=sr. (1)

This is also extended to the exradii

~=(s-a)ra=(s-b )rb=(s-c)rco (2) Both these equations are fundamental in deriving the formula for area of a triangle in relation to sides of the triangle and the semi-perimeter

~=

,J'--s(-s -a )-( s-b-)(-s -c-)

Heron's formula is useful in one of the proofs of Descartes circle theorem of the curvatures of mutually tangent circles that:

Given the radii and curvatures of any four tangent circles E1, E2, E^3, E4 . are fi and €i (i=1,2,3 and 4) respectively then

2(E12 + E22 + E32 +E42)

=

Descartes theorem is proved and is used to show that to every three mutually tangent circles there are two other non- intersecting circles tangent to all three (see Fig.l). The formula for finding the radii of these circles, called the inner and outer Soddy circles, is thus derived

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where

A

~r1r2r3 (r1 + r2 + r3)

and the negative sign applies to the outer Soddy circle.

Fig.]

Given also that the inner and outer Soddy circles have curvatures €4 and €'4 respectively, it is proved that:

E4 - E' 4 =4114, (7)

where 114 is the curvature of the incircle of triangle ~C, and A, B and C are the centers ofE}, E2 and E3 respectively.

San Gaku is Japanese traditional geometry, which was practiced in different prefectures at differ:ent times during Japan's period of isolation from the rest of the world. Tablets of geometry problems included many tangent circles.

Some of these tangent circle problems are discussed in this paper. Conjectures are made and then proved by induction or

otherwise. In others an application of Descartes circle theorem has been a useful tool to prove the conjectures.

Fig. 2

For example for n=3, 4, .... and r the radius of each of the bigger circles in fig.2, the set of tangent circles indicated above as with radii decreasing in the direction of

i) A are given by the formula

~_ (n~I)2

(8)

ii) B are given by the formula

1 2(n-I)(n-2)

rn ^r (9)

iii) C are given by the formula

1 (Fn)2

(10)

rn r

where Fn is the nth Fibonacci number.

Case (iii) is not in San Gaku and each subsequent circle Cn is tangent to the two previous circles Cn-1 and Cn-2 and to the straight line AC.

These theorems are proved by induction.

Many more San Gaku problems have been proved and the application of these to the study of curvatures leads to an interesting study in number theory.

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For example from equation (8), if r=1 then the sequence of curvatures generated from this formula including the two circles of radius 1, is

1,1,4,9,16,25, ... .

which are perfect squares while from (9) we have (for r=1) the sequence

1,1,4,12,24,40,60,84, ...

What is interesting is that if the first three curvatures are integral then all the circles in the chain are also all integral For example, given that the four similar symmetric spirals of tangent circles in fig.3 start with circles of radii 1 , 1/2 and 1/ 2, then the curvatures of the rest are given by

~=2+n2

rn

and the sequence of these beginning with the outer one is

-1, 2, 2, 3, 6, 11, 18,27, ....

The paper concludes with a brief study in

cyclotomy