Dear Nikos and Tuan

Thanks for your interesting remarks.

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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