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Tangent Circles


TangentCircles

Two circles with centers at (x_i,y_i) with radii r_i for i=1,2 are mutually tangent if

 (x_1-x_2)^2+(y_1-y_2)^2=(r_1+/-r_2)^2.
(1)

If the center of the second circle is inside the first, then the - and + signs both correspond to internally tangent circles. If the center of the second circle is outside the first, then the - sign corresponds to externally tangent circles and the + sign to internally tangent circles.

Finding the circles tangent to three given circles is known as Apollonius' problem. The Desborough Mirror, a beautiful bronze mirror made during the Iron Age between 50 BC and 50 AD, consists of arcs of circles that are exactly tangent (Wolfram 2002, pp. 43 and 873).

TangentCirclesAtTriangleVertices

Given three distinct noncollinear points A, B, and C, denote the side lengths of the triangle DeltaABC as a, b, and c. Now let three circles be drawn, one centered about each point and each one tangent to the other two (left figure), and call the radii r_1=a^', r_2=b^', r_3=c^'.

Interestingly, the pairwise external similitude centers of these circles are the three Nobbs points (P. Moses, pers. comm., Mar. 14, 2005).

These three circles satisfy

a^'+b^'=c
(2)
a^'+c^'=b
(3)
b^'+c^'=a
(4)

(right figure). Solving for the radii gives

a^'=1/2(-a+b+c)
(5)
b^'=1/2(a-b+c)
(6)
c^'=1/2(a+b-c).
(7)

Plugging these equations in to the equation of the semiperimeter of DeltaABC

 s=1/2(a+b+c).
(8)

gives

 2s=(a^'+b^')+(a^'+c^')+(b^'+c^')=2(a^'+b^'+c^'),
(9)

so

 a^'+b^'+c^'=s.
(10)

In addition,

 a=b^'+c^'=a^'+b^'+c^'-a^'=s-a^'.
(11)

Switching a and a^' to opposite sides of the equation and noting that the above argument applies equally well to b^' and c^' then gives the radii of the three circles as

a^'=s-a
(12)
b^'=s-b
(13)
c^'=s-c.
(14)

The pairwise points of tangency of the three circles are precisely the vertices of the contact triangle DeltaC_AC_BC_C of DeltaABC, i.e., the triangle formed by the points at which the incircle is tangent to the original triangle. The circles that are internally and externally tangent to these three circles are known as the Soddy circles.

No Kimberling centers lie on any of the tangent circles.

The radical circle of the tangent circles is the incircle.

Two circles of radii r_1 and r_2 with centers separated by a distance d are externally tangent if

 d=r_1+r_2
(15)

and internally tangent if

 d=|r_1-r_2|.
(16)
TangentCirclesFeuerbachPoint

The following table summarizes tangent circles for some common named circles. As can be seen, the incircle, nine-point circle, and Moses circle are mutually tangent at the Feuerbach point.

TangentCirclesTriangle

There are four circles that are tangent all three sides (or their extensions) of a given triangle: the incircle I and three excircles J_1, J_2, and J_3. These four circles are, in turn, all touched by the nine-point circle N.

TangentCirclesOnALine

If two circles C_1 and C_2 of radii r_1 and r_2 are mutually tangent to each other and a line, then their centers are separated by a horizontal distance given by solving

 x_2^2+(r_1-r_2)^2=(r_1+r_2)^2
(17)

for x_2, giving

 x_2=2sqrt(r_1r_2).
(18)

The position and radius of a third circle tangent to the first two and the line can be found by solving the simultaneous equations

x_3^2+(r_1-r_3)^2=(r_1+r_3)^2
(19)
(x_3-x_2)^2+(r_2-r_3)^2=(r_2+r_3)^2
(20)

for x_3 and r_3, giving

x_3=(2r_1sqrt(r_2))/(sqrt(r_1)+sqrt(r_2))
(21)
r_3=(r_1r_2)/((sqrt(r_1)+sqrt(r_2))^2).
(22)

The latter equation can be written in the form

 1/(sqrt(r_3))=1/(sqrt(r_1))+1/(sqrt(r_2)).
(23)

This problem was given as a Japanese temple problem on a tablet from 1824 in the Gumma Prefecture (Rothman 1998).


See also

Apollonius' Problem, Casey's Theorem, Chain of Circles, Circle Packing, Circle-Circle Tangents, Descartes Circle Theorem, Excircles, Four Coins Problem, Hawaiian Earring, Incircle, Inner Soddy Circle, Lens, Lune, Malfatti Circles, Malfatti's Problem, Nine Circles Theorem, Outer Soddy Circle, Oval, Pappus Chain, Seven Circles Theorem, Six Circles Theorem, Soddy Circles, Tangent Curves, Tangent Spheres

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References

Coolidge, J. L. "Mutually Tangent Circles." §1.3 in A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 31-44, 1971.Fukagawa, H. and Pedoe, D. "Two Circles," "Three Circles," "Four Circles," and "Many Circles." §1.1-1.5 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 3-13 and 79-88, 1989.Hannachi, N. "Kissing Circles." http://perso.wanadoo.fr/math-a-mater/pack/packingcircle.htm.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 43 and 873, 2002.

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Tangent Circles

Cite this as:

Weisstein, Eric W. "Tangent Circles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TangentCircles.html

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