## 15393Re: [EMHL] Re: Cevian Trace Equal Area Points

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• Aug 1, 2007
Dear Nikos and Tuan
Of course in choo-choo theory, these lines La, Lb, Lc play a central role
and I have called them equal area axis.
Notice that their construction is affine! I had also my own construction
slightly different from yours.
As for Tuan formula, I would be happy to have a name if any for the dual
point M(q*w - r*v::)of the line through P(p:q:r) and Q(u:v:w) given by their
barycentrics.
Friendly
Francois
PS As Nikos notice, point isotomic of the areal center also plays an
important role in choo-choo theory.
As I go away from Paris in Britanny for several weeks even months far away
from the web, I give you a choo-choo configuration, so you can think about
me in this summer time:
Instead of cevian tiangles, I will look at pedal triangles PaPbPc and QaQbQc
of points P and Q wrt triangle ABC and I call L the line through P and Q.
Let E be the equicenter and S the areal center of the pedal triangles.
Then:
1° E is the orthopole of line L wrt ABC.
2° S is on the ABC-circumcircle and its Simson line is parallel to line L.

The first point was knew by Neuberg for a very very long time and maybe
that's why he found the orthopole. The first proof I saw was due to
V.Thebault
As for the second point, I would be happy to have some reference if any.
Of course these properties of points E and S are shared by any pair of
triangles of points on line L but this is another (choo-choo) story.

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