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1Lemoine point

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  • Clark Kimberling
    Dec 22, 1999
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      Dear triangle geometers,

      I'm sure I speak for many who thank Antreas for establishing
      Hyacinthos@onelist.

      Antreas mentioned that that name Hyacinthos honors E. Lemoine, of whose
      full name Hyacinthe is a part. The Lemoine point is often called the
      symmedian point. In Ross Honsberger's Episodes in Nineteenth and Twentieth
      Century Euclidean Geometry (Mathematical Association of America, 1995), a
      whole chapter is devoled to this point.

      However, Honsberger doesn't mention (directly) a certain interesting
      property of the Lemoine point. For any point P, let A'B'C' denote the
      pedal triangle of P (i.e., A' is the point in which the line through P
      perpendicular to line BC meets line BC). Let S(P) be the vector sum
      PA'+PB'+PC'. Then S(P) is the zero vector if P is the Lemoine point.

      I conjecture that the converse is true: that if P is a "point" (i.e.,
      f(a,b,c) : g(a,b,c) : h(a,b,c)) such that S(P)=0, then P = a^2 : b^2 : c^2
      (barycentric coordinates of the Lemoine point).

      By the way, many other vector sums involving triangle centers will be
      included in ETC (Encyclopedia of Triangle Centers), which should appear
      sometime before March 1, 2000.

      Best holiday regards to all.

      Clark Kimberling
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