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Difference between revisions of "Descartes' Circle Formula"

(Created page with '(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 circl…')
 
 
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(based on wording of ARML 2010 Power)
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Descartes' Circle Formula is a relation held between four mutually tangent circles.
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==Definition of Curvature==
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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>.
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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.
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Descartes' Circle Formula is a relation held between four mutually tangent circles.  
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<asy>
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size(200);
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defaultpen(linewidth(0.7));
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draw(Circle(origin,0.5));
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draw(Circle((1.5,0),1));
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dot(origin^^(1.5,0)^^(0.5,0));
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draw(origin--(1.5,0));
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label("$1/2$", (0.25,0), N);
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label("$1$", (1,0), N);
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label("$A$", origin, SW);
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label("$B$", (1.5,0), SE);
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</asy>
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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>.
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<asy>
 +
size(150);
 +
defaultpen(linewidth(0.7));
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draw(Circle((1.25,0),0.25));
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draw(Circle((1.5,0),0.5));
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dot((1,0)^^(1.5,0)^^(1.25,0)^^(2,0));
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draw((1,0)--(2,0));
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label("$1/2$", (1.125,0), N);
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label("$1$", (1.75,0), N);
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label("$A$", (1.25,0), SW);
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label("$B$", (1.5,0), SE);
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</asy>
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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>.
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==Statement==
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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:
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<math>(r_1 + r_2 + r_3 + r_4)^2 = 2(r_1^2 + r_2^2 + r_3^2 + r_4^2)</math>.
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==Proof==
  
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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_1}</math> and <math>\frac{1}{r_2}</math>.
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Waiting for the MAA to grant permission to avoid copyright issues...
  
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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.
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==Problems==
  
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[asy]
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[[Category:Theorems]]
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draw(Circle(origin,2));
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[[Category:Geometry]]
−
[/asy]
 

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 $r_a$ 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 $-1$.

Statement

When four circles are pairwise tangent, with respective curvatures $r_1, r_2, r_3,$ and $r_4$, then the following equation holds:

$(r_1 + r_2 + r_3 + r_4)^2 = 2(r_1^2 + r_2^2 + r_3^2 + r_4^2)$.

Proof

Waiting for the MAA to grant permission to avoid copyright issues...

Problems

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