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Casey's Theorem


Four circles c_1, c_2, c_3, and c_4 are tangent to a fifth circle or a straight line iff

 T_(12)T_(34)+/-T_(13)T_(42)+/-T_(14)T_(23)=0.
(1)

where T_(ij) is the length of a common tangent to circles i and j (Johnson 1929, pp. 121-122). The following cases are possible:

1. If all the Ts are direct common tangents, then c_5 has like contact with all the circles,

2. If the Ts from one circle are transverse while the other three are direct, then this one circle has contact with c_5 unlike that of the other three,

3. If the given circles can be so paired that the common tangents to the circles of each pair are direct, while the other four are transverse, then the members of each pair have like contact with c_5

(Johnson 1929, p. 125).

CaseysTheorem

The special case of Casey's theorem shown above was given in a Sangaku problem from 1874 in the Gumma Prefecture. In this form, a single circle is drawn inside a square, and four circles are then drawn around it, each of which is tangent to the square on two of its sides. For a square of side length a with lower left corner at (0,0) containing a central circle of radius r with center (x,y), the radii and positions of the four circles can be found by solving

 (1-r_4-x)^2+(y-r_4)^2=(r+r_4)^2
(2)
 (1-r_1-x)^2+(1-r_1-y)^2=(r+r_1)^2
(3)
 (x-r_3)^2+(y-r_3)^2=(r+r_3)^2
(4)
 (x-r_2)^2+(1-r_2-y)^2=(r+r_2)^2.
(5)

Four of the T_(ij) for the theorem are given immediately for the figure as

T_(12)=a-r_1-r_2
(6)
T_(34)=a-r_3-r_4
(7)
T_(14)=a-r_1-r_4
(8)
T_(23)=a-r_2-r_3.
(9)

The remaining T_(13) and T_(24) can be found as shown in the above right figure. Let c_(ij) be the distance from O_i to O_j, then

c_(13)^2=(a-r_1-r_3)^2+(a-r_1-r_3)^2
(10)
=2(a-r_1-r_3)^2
(11)
c_(24)^2=(a-r_2-r_4)^2+(a-r_2-r_4)^2
(12)
=2(a-r_2-r_4)^2,
(13)

so

T_(13)=sqrt(c_(13)^2-(r_3-r_1)^2)
(14)
=sqrt(2(a-r_1-r_3)^2-(r_3-r_1)^2)
(15)
T_(24)=sqrt(c_(24)^2-(r_2-r_4)^2)
(16)
=sqrt(2(a-r_2-r_4)^2-(r_2-r_4)^2).
(17)

Since the four circles are all externally tangent to c_5, the relevant form of Casey's theorem to use has signs (+,-), so we have the equation

 (a-r_1-r_2)(a-r_3-r_4)+(a-r_1-r_4)(a-r_2-r_3) 
 -sqrt([2(a-r_1-r_3)^2-(r_3-r_1)^2][2(a-r_2-r_4)^2-(r_2-r_4)^2])=0
(18)

(Rothman 1998). Solving for a then gives the relationship

 a=(2(r_1r_3-r_2r_4)+sqrt(2(r_1-r_2)(r_1-r_4)(r_3-r_2)(r_3-r_4)))/(r_1-r_2+r_3-r_4)
(19)

Durell (1928) calls the following Casey's theorem: if t is the length of a common tangent of two circles of radii a and b, t^' is the length of the corresponding common tangent of their inverses with respect to any point, and a^' and b^' are the radii of their inverses, then

 (t^2)/(ab)=(t^('2))/(a^'b^').
(20)

See also

Purser's Theorem, Tangent Circles

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References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 103, 1888.Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 125, 1893.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 37, 1971.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928.Fukagawa, H. and Pedoe, D. "Many Circles and Squares (Casey's Theorem)." §3.3 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 41-42 and 120-1989.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 121-127, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 244-251, 1893.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.

Referenced on Wolfram|Alpha

Casey's Theorem

Cite this as:

Weisstein, Eric W. "Casey's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CaseysTheorem.html

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