Difference between revisions of "Descartes' Circle Formula"
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| − | + | Descartes' Circle Formula is a relation held between four mutually tangent circles. | |
| − | + | ==Definition of Curvature== | |
| + | 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_a}</math> and <math>\frac{1}{r_b}</math>. | ||
| − | + | 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_a}</math>, while the curvature of circle B is <math>-\frac{1}{r_b}</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>. |
| + | |||
| + | ==Statement== | ||
| + | When four circles are pairwise tangent, with respective curvatures <math>r_1, r_2, r_3,</math> and <math>r_4</math>, then the following equation holds: | ||
| + | |||
| + | <math>(r_1 + r_2 + r_3 + r_4)^2 = 2(r_1^2 + r_2^2 + r_3^2 + r_4^2)</math>. | ||
| + | |||
| + | ==Proof== | ||
| + | |||
| + | Waiting for the MAA to grant permission to avoid copyright issues... | ||
| − | + | ==Problems== | |
| − | + | [[Category:Theorems]] | |
| + | [[Category:Geometry]] | ||
Latest revision as of 16:22, 22 August 2025
Descartes' Circle Formula is a relation held between four mutually tangent circles.
Definition of Curvature
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 
.
Statement
When four circles are pairwise tangent, with respective curvatures 
and 
, then the following equation holds:
.
Proof
Waiting for the MAA to grant permission to avoid copyright issues...
