View source for Descartes' Circle Formula

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>
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 <math>A</math> is <math>2</math> while the curvature of circle <math>B</math> is <math>1</math>.

<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 <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==

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==Problems==

[[Category:Theorems]]
[[Category:Geometry]]