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