On 2 May Darij asked how the converse of Casey's

Theorem can be proved.

R A Johnson,'Advanced Euclidean Geometry' Dover 1960

reprint addresses Casey's Theorem(p.122) which he says

was first given by Casey in incomplete form. He also

says that the more important converse has frequently

been proved under various restrictions, but he does

not elaborate.

He refers to R Lachlan,'Modern Pure Geometry'

Macmillan, London 1893 who addresses a 'System of four

circles having a common tangent circle' (p.244-250).

He quotes from a paper by A Larmor, Proc LMS 1891

which "shows that the converse is true under all

conditions". Because of the need to consider various

conditions of direct and transverse common tangents,

and point circles, and because of references made to

lemmas, etc proven elsewhere in the book, I do not

feel competent to summarise the proof.

Lachlan also outlines a second proof suggested by H F

Baker. Again the various conditions of the tangents

and circles must be considered "by which the truth of

the theorem may be inferred" and "the general case may

be deduced".

I hope these references will help anyone keen to

follow the details of this theorem and its converse.

Regards, Peter Scales.

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