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Difference between revisions of "Descartes' Circle Formula"

 
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Descartes' Circle Formula is a relation held between four mutually tangent circles.  
 
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 <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>.
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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_a}</math> and <math>\frac{1}{r_b}</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.
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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>
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</asy>
 
</asy>
  
In the above diagram, the curvature of circle A is 2 while the curvature of circle B is 1.
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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(200);
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size(150);
 
defaultpen(linewidth(0.7));
 
defaultpen(linewidth(0.7));
 
draw(Circle((1.25,0),0.25));
 
draw(Circle((1.25,0),0.25));
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</asy>
 
</asy>
  
In the above diagram, the curvature of circle A is still 2 while the curvature of circle B is -1.
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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:
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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>.
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[[Category:Theorems]]
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[[Category:Geometry]]

Latest revision as of 20:00, 24 December 2017

(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 $r_a$ is externally tangent to circle B of radius $r_b$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_a}$ and $\frac{1}{r_b}$.

If circle $A$ is internally tangent to circle $B$, however, a the curvature of circle $A$ is still $\frac{1}{r_a}$, while the curvature of circle B is $-\frac{1}{r_b}$, 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]

In the above diagram, the curvature of circle $A$ is $2$ while the curvature of circle $B$ is $1$.

[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]

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 the following equation holds:

$(a+b+c+d)^2 = 2(a^2+b^2+c^2+d^2)$.

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