Difference between revisions of "Descartes' Circle Formula"
| Line 5: | Line 5: | ||
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 <math>r_a</math> is externally tangent to circle B of radius <math>r_b</math>. Then the curvatures of the circles are simply the reciprocals of their radii, <math>\frac{1}{r_1}</math> and <math>\frac{1}{r_2}</math>. | 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 <math>r_a</math> is externally tangent to circle B of radius <math>r_b</math>. Then the curvatures of the circles are simply the reciprocals of their radii, <math>\frac{1}{r_1}</math> and <math>\frac{1}{r_2}</math>. | ||
| − | 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 <math>A</math> is internally tangent to circle <math>B</math>, however, a the curvature of circle <math>A</math> 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> | <asy> | ||
| Line 20: | Line 20: | ||
</asy> | </asy> | ||
| − | In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1. | + | In the above diagram, the curvature of circle <math>A</math> is <math>2</math> while the curvature of circle <math>B</math> is <math>1</math>. |
<asy> | <asy> | ||
| − | size( | + | size(150); |
defaultpen(linewidth(0.7)); | defaultpen(linewidth(0.7)); | ||
draw(Circle((1.25,0),0.25)); | draw(Circle((1.25,0),0.25)); | ||
| Line 35: | Line 35: | ||
</asy> | </asy> | ||
| − | In the above diagram, the curvature of circle A is still 2 while the curvature of circle B is -1. | + | In the above diagram, the curvature of circle <math>A</math> is still <math>2</math> while the curvature of circle <math>B</math> is <math>-1</math>. |
| − | When four circles A, B, C, and D are pairwise tangent, with respective curvatures a, b, c, and d, then: | + | When four circles <math>A, B, C,</math> and <math>D</math> are pairwise tangent, with respective curvatures <math>a, b, c,</math> and <math>d</math>, then the following equation holds: |
<math>(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)</math>. | <math>(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)</math>. | ||
Revision as of 22:45, 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 
is internally tangent to circle 
, however, a the curvature of circle 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 is 
while the curvature of circle is 
.
![[asy] size(150); 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/f/9/4/f94615bc59e3ebcdc39715aa9864b8a03a34440e.png)
In the above diagram, the curvature of circle is still while the curvature of circle is 
.
When four circles 
and 
are pairwise tangent, with respective curvatures 
and 
, then the following equation holds:
.
