Loading ...
Sorry, an error occurred while loading the content.

Casey question

Expand Messages
  • Darij Grinberg
    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
    Message 1 of 3 , May 2, 2003
    • 0 Attachment
      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.

      Darij
    • caiser@cantv.net
      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
      Message 2 of 3 , May 3, 2003
      • 0 Attachment
        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
        2/r=1/r1+1/r2=1/r3+1/r4=1/r5+1/r6
        Is this very knowked?
        Thanks
        Juan Carlos
      • Peter Scales
        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
        Message 3 of 3 , May 7, 2003
        • 0 Attachment
          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.

          http://mobile.yahoo.com.au - Yahoo! Mobile
          - Check & compose your email via SMS on your Telstra or Vodafone mobile.
        Your message has been successfully submitted and would be delivered to recipients shortly.