Four circles , , , and are tangent to a fifth circle or a straight line iff
(1)
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where is the length of a common tangent to circles and (Johnson 1929, pp. 121-122). The following cases are possible:
1. If all the s are direct common tangents, then has like contact with all the circles,
2. If the s from one circle are transverse while the other three are direct, then this one circle has contact with 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
(Johnson 1929, p. 125).
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 with lower left corner at containing a central circle of radius with center , the radii and positions of the four circles can be found by solving
(2)
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(3)
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(4)
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(5)
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Four of the for the theorem are given immediately for the figure as
(6)
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(7)
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(8)
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(9)
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The remaining and can be found as shown in the above right figure. Let be the distance from to , then
(10)
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(11)
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(12)
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(13)
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so
(14)
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(15)
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(16)
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(17)
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Since the four circles are all externally tangent to , the relevant form of Casey's theorem to use has signs , so we have the equation
(18)
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(Rothman 1998). Solving for then gives the relationship
(19)
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Durell (1928) calls the following Casey's theorem: if is the length of a common tangent of two circles of radii and , is the length of the corresponding common tangent of their inverses with respect to any point, and and are the radii of their inverses, then
(20)
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