- Dec 22, 1999Dear triangle geometers,
I'm sure I speak for many who thank Antreas for establishing
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.
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