Difference between revisions of "Descartes' Circle Formula"
m (moved Descartes' circle formula to Descartes' Circle Formula: Capitalizing Title) |
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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> | |
− | draw(Circle(origin,2)); | + | 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>. | ||
+ | |||
+ | 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>. | ||
+ | |||
+ | [[Category:Theorems]] | ||
+ | [[Category:Geometry]] |
Latest revision as of 21:00, 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:
.