# 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}{ | + | 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}{ | + | 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. | + | 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)); | ||

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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. | + | 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>. | ||

+ | |||

+ | [[Category:Theorems]] | ||

+ | [[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 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.

In the above diagram, the curvature of circle is while the curvature of circle is .

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:

.