## Friday, January 1, 2010

### Concept 1 :KiSsInG cIrClEs

Descartes' theorem (1964)
If four mutually tangent circles have curvature ki (for i = 1,...,4), (ki=1/ri, Where ri is radius)
 (1)
$(k_1+k_2+k_3+k_4)^2=2\,(k_1^2+k_2^2+k_3^2+k_4^2).$
When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:
 (2)
$k_4 = k_1 + k_2 + k_3 \pm2 \sqrt{k_1 k_2 + k_2 k_3 + k_3 k_1}.$
The ± sign reflects the fact that there are in general two solutions. Ignoring the degenerate case of a straight line, one solution is positive and the other is either positive or negative; if negative, it represents a circle that circumscribes the first three (as shown in the diagram above).
Other criteria may favor one solution over the other in any given problem.

Special case
If one of the three circles is replaced by a straight line, then one ki, say k3, is zero and drops out of equation (1). Equation (2) then becomes much simpler:
 (3)
$k_4=k_1+k_2\pm2\sqrt{k_1k_2}.$
Example:
If three circles of radius 1 touches one another then find the radius of the circle which can be inserted within these circles, touching all the three.[CAT]

Solution:
k4=3+2root(3)[positive for inscribed circle] therefore radius=1/[3+2root(3)]=[2-root(3)]/root(3)

PS:One can use conventional method too.