# Casey's Theorem

Casey's theorem is a generalization of Ptolemy's. The latter deals with an inscribed quadrilateral:

Let a convex quadrilateral ABCD be inscribed in a circle. Then the sum of the products of the two pairs of opposite sides equals the product of its two diagonals. In other words,

Casey's generalization claims a similar identity for the common tangents of four circles tangent to a fifth. (Ptolemy's case is obtained by letting all four circles have 0 radius.)

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Casey proved his theorem in the 1881 edition of his book. Coolidge made on everbearing reference to Casey's treatment [Coolidge, p. 37, a footnote]:

See his greatly overrated A Sequel to Euclid, 1881, p. 101. The ingenious writer makes two characteristic mistakes. He assumes that in proving the theorem he has also proved the converse. Secondly, he omits to require his circles to be mutually external. But in that case it is easy to find four circles tangent to a fifith whereof one surrounds the three others and has no common tangential segments with them, in the ral domain.

[Johnson, p. 122] also observes the absence of the converse statement in Casey's formulation but then writes in a footnote (p. 124):

It does not appear that most of the writers have stated the exact limitations on the validity of this theorem. Casey, Lachlan, and others, do not sufficiently restrict it; while Coolidge unnecessarily limits its scope by stipulating that the circles be mutually external. It will be evident that the theorem as here given includes all the cases where the tangents in question exist; in other words, all the cases in which the formula can be stated in terms of real numbers.

In the 1888 edition of Casey's book the theorem appears on p. 103:

If four circles be all touched by the same circle; then, denoting 12, the common tangent of the 1st and 2nd, &c.,

12 · 34 + 14 · 23 = 13 · 24.

The statement comes with a foonote remark that the author had it first published in 1866 in the Proceedings of the Royal Irish Academy.

I must say that Coolidge's critique is not entirely justfied. Ptolemy himself did not need the converse theorem for his astronomic pursuits. He proved the direct statement and left it to us to guess whether he was aware that the converse is also true. Thus Casey was within his rights to claim an extension to Ptolemy's theorem.

[Fukagawa & Pedoe, p. 120] write (perhaps mistakenly) that the theorem has been stated by John Casey in 1857. They continue:

There is no doubt about the necessity of this condition, but the sufficiency is dependent on other given conditions. The necessity was stated by Chochu Siraisi in 1830.

From later publications, it appears that Chochu Siraisi expected all four circles touch the fifth either internally or externally, so that all the tangents in the formula are external. Casey, however, immediately after proving his theorem, derives, as a corollary, the existence of Feuerbach's circle as the circle tangent to the incircle of a triangle internally and to the three excircles externally.

... to be continued ...

### References

1. J. Casey, A Sequel to Euclid, Scholarly Publishing Office, University of Michigan Library (December 20, 2005), reprint of the 1888 edition
2. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
3. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
4. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960