- Does anybody know how the converse of the Casey theorem can be
proven? The theorem itself is shown in several places like
http://www.pandd.demon.nl/casey.htm but it is the converse that
makes it so helpful.
- Dear Darij and other friends:
With respect to Casey`s theorem application. I finded
some relations for a triangle with these conditions:
Let ABC triangle with incircle(I) with radius r and
circumcircle(O),be BBb bisector of <B with pedal point
Bb on AC side,also the inscribed circles (O1) with
radius r1 and (O2) with radius r2 in the
mixtilinear triangles conformed by arc AB of circle O
(opposed to C vertex) ,ABb, BBb and the other by arc BC
of circle O(opposed to A vertex),BBb,CBb respectively.
Similarly we obtain other four circles r3 and r4 for AAa
bisector and r5 and r6 for CCc bisector.Then:
6/r=1/r1+1/r2+1/r3+1/r4+1/r5+1/r6 and of course
Is this very knowked?
- 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
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
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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