Difference between revisions of "Descartes' Circle Formula"
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If circle A is internally tangent to circle B, however, a the curvature of circle A is still <math>\frac{1}{r_1}</math>, while the curvature of circle B is <math>-\frac{1}{r_2}</math>, the opposite of the reciprocal of its radius. | If circle A is internally tangent to circle B, however, a the curvature of circle A is still <math>\frac{1}{r_1}</math>, while the curvature of circle B is <math>-\frac{1}{r_2}</math>, the opposite of the reciprocal of its radius. | ||
| − | + | <asy> | |
| − | draw(Circle(origin,2)); | + | size(200); |
| − | + | defaultpen(linewidth(0.7)); | |
| + | draw(Circle(origin,0.5)); | ||
| + | draw(Circle((1.5,0),1)); | ||
| + | dot(origin^^(1.5,0)^^(0.5,0)); | ||
| + | draw(origin--(1.5,0)); | ||
| + | label("$1/2$", (0.25,0), N); | ||
| + | label("$1$", (1,0), N); | ||
| + | label("$A$", origin, SW); | ||
| + | label("$B$", (1.5,0), SE); | ||
| + | </asy> | ||
| + | |||
| + | In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1. | ||
| + | |||
| + | <asy> | ||
| + | size(200); | ||
| + | defaultpen(linewidth(0.7)); | ||
| + | draw(Circle((1.25,0),0.25)); | ||
| + | draw(Circle((1.5,0),0.5)); | ||
| + | dot((1,0)^^(1.5,0)^^(1.25,0)^^(2,0)); | ||
| + | draw((1,0)--(2,0)); | ||
| + | label("$1/2$", (1.125,0), N); | ||
| + | label("$1$", (1.75,0), N); | ||
| + | label("$A$", (1.25,0), SW); | ||
| + | label("$B$", (1.5,0), SE); | ||
| + | </asy> | ||
| + | |||
| + | In the above diagram, the curvature of circle A is still 2 while the curvature of circle B is -1. | ||
| + | |||
| + | When four circles A, B, C, and D are pairwise tangent, with respective curvatures a, b, c, and d, then: | ||
| + | |||
| + | <math>(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)</math>. | ||
Revision as of 23:43, 11 March 2011
(based on wording of ARML 2010 Power)
Descartes' Circle Formula is a relation held between four mutually tangent circles.
Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle A of radius 
is externally tangent to circle B of radius 
. Then the curvatures of the circles are simply the reciprocals of their radii, 
and 
.
If circle A is internally tangent to circle B, however, a the curvature of circle A is still , while the curvature of circle B is 
, the opposite of the reciprocal of its radius.
![[asy] size(200); defaultpen(linewidth(0.7)); draw(Circle(origin,0.5)); draw(Circle((1.5,0),1)); dot(origin^^(1.5,0)^^(0.5,0)); draw(origin--(1.5,0)); label("$1/2$", (0.25,0), N); label("$1$", (1,0), N); label("$A$", origin, SW); label("$B$", (1.5,0), SE); [/asy]](http://latex.artofproblemsolving.com/a/1/0/a10298bb91d00b0aaace608317f8d041d7040165.png)
In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1.
![[asy] size(200); defaultpen(linewidth(0.7)); draw(Circle((1.25,0),0.25)); draw(Circle((1.5,0),0.5)); dot((1,0)^^(1.5,0)^^(1.25,0)^^(2,0)); draw((1,0)--(2,0)); label("$1/2$", (1.125,0), N); label("$1$", (1.75,0), N); label("$A$", (1.25,0), SW); label("$B$", (1.5,0), SE); [/asy]](http://latex.artofproblemsolving.com/a/f/4/af47ea9e3ec1c5e986ebbc66c01ff3e06addce01.png)
In the above diagram, the curvature of circle A is still 2 while the curvature of circle B is -1.
When four circles A, B, C, and D are pairwise tangent, with respective curvatures a, b, c, and d, then:
.
