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Unlikey concurrences

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  • Wilson Stothers
    To Jean-Louis Ayme and all at Hyacinthos Please forgive the last abortive post. It is almost two years since the original post, but there is something to add
    Message 1 of 2 , Sep 3, 2005
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      To Jean-Louis Ayme and all at Hyacinthos

      Please forgive the last abortive post.

      It is almost two years since the original post, but there is
      something to add

      Here is the original relating to Ayme's paper in Crux Math

      We have the following concurrences;

      1. (interior) angle-bisector of A.
      2. Perpendicular to 1. through B
      3. Join of mid-points of AB, BC.
      4. Join of touches of incircle with BC, CA.

      1. Symmedian through A.
      2. Join of feet of altitudes from A, B.
      3. As 3. above.
      4. Parallel thru B to tan at A to circumcircle.

      In fact, there is a fifth line

      Let T(X) denote the tripolar of a point X

      THEOREM
      Suppose that
      A'B'C' is the cevian triangle of P
      A"B"C" is the cevian triangle of Q
      A*B*C* is the cevian triangle of R
      where R is the crosspoint of P nd Q.
      Then the following FIVE lines are concurrent
      C'A'
      B"A"
      AR
      BX
      CY,
      where
      X = B*C*nT(P)
      Y = B*C*nT(Q)

      Ayme's examples are

      P = G, X = X(7), so R = X(1),
      P = G, X = X(4), so R = X(6).

      We might add

      P = G, X = X(8), so R = X(9),
      P = G, X = X(69), so R = X(3).

      So far, B' and C" are left out - has anyone an idea how to include
      them?

      Best wishes

      Wilson
    • Wilson Stothers
      Dear All Sorry - there is a slip in the previous message A*B*C* is the ANTIcevian triangle of R Wilson
      Message 2 of 2 , Sep 3, 2005
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        Dear All

        Sorry - there is a slip in the previous message

        A*B*C* is the ANTIcevian triangle of R

        Wilson

        --- In Hyacinthos@yahoogroups.com, "Wilson Stothers" <wws@m...> wrote:
        > To Jean-Louis Ayme and all at Hyacinthos
        >
        > Please forgive the last abortive post.
        >
        > It is almost two years since the original post, but there is
        > something to add
        >
        > Here is the original relating to Ayme's paper in Crux Math
        >
        > We have the following concurrences;
        >
        > 1. (interior) angle-bisector of A.
        > 2. Perpendicular to 1. through B
        > 3. Join of mid-points of AB, BC.
        > 4. Join of touches of incircle with BC, CA.
        >
        > 1. Symmedian through A.
        > 2. Join of feet of altitudes from A, B.
        > 3. As 3. above.
        > 4. Parallel thru B to tan at A to circumcircle.
        >
        > In fact, there is a fifth line
        >
        > Let T(X) denote the tripolar of a point X
        >
        > THEOREM
        > Suppose that
        > A'B'C' is the cevian triangle of P
        > A"B"C" is the cevian triangle of Q

        > A*B*C* is the cevian triangle of R *********************

        > where R is the crosspoint of P nd Q.
        > Then the following FIVE lines are concurrent
        > C'A'
        > B"A"
        > AR
        > BX
        > CY,
        > where
        > X = B*C*nT(P)
        > Y = B*C*nT(Q)
        >
        > Ayme's examples are
        >
        > P = G, X = X(7), so R = X(1),
        > P = G, X = X(4), so R = X(6).
        >
        > We might add
        >
        > P = G, X = X(8), so R = X(9),
        > P = G, X = X(69), so R = X(3).
        >
        > So far, B' and C" are left out - has anyone an idea how to include
        > them?
        >
        > Best wishes
        >
        > Wilson
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