Dear bbblow

All I can say is you trying to search on all my past posts in Hyacinthos.

Three points moving with equal speed on three given lines is just what I

have called chowchow, remembering the noise of our old steam trains rolling

on their tracks.

A chowchow have in general 2 centers: the areal center and the equicenter.

It is Neuberg who found the equicenter at the turn of the 20th century.

In the other way, given the areal center and the equicenter, you get a

unique chowchow, up to an affine change of the law of time.

So a chowchow, your trisector configuration have its two centers: the areal

center is I the incenter and the equicenter is J the symmetric of I wrt the

centroid G.

Now , I give you a funny construction of your triples (D, E, F).

In fact, I construct a special triple (Do, Eo, Fo) in this way.

Let O be the circumcenter and O' be the point of the line IO such that the

affine ratio: OO'/OI = -1/3

Then the triangle DoEoFo is the pedal triangle of O' wrt the triangle of

reference ABC.

Now you get any other triple (D, E, F) since the signed segments DoD = EoE =

FoF are equal.

Friendly

Francois

For a general chowchow, the trace of the circumcenter of the triple (D, E,

F) is no longer a conic!

On Sat, Sep 18, 2010 at 2:25 PM, bbblow <bbbbbblow@...> wrote:

>

>

> Actually the property does not only hold for points trisecting a triangle

> perimeter, but also works in the following generalized form:

>

> On three given lines, three points move in equal speed. Then the

> circumcenters of the three points trace out a conic.

>

> An important point to be taken is that if the three points are moving in

> different speeds then the locus is not a conic. (All these were verified

> using GSP)

>

> Possible further studies would be:

> 1. If we put n lines in n dimension, and animate n points on each line,

> what would the hyper-circumcenter trace out?

> 2. What determines the type of the conic?

> 3. How do we 'standardize' this property to barycentric coordinates? (Three

> lines mean three intersection points, and hence triangle geometry would be

> relevant)

> 4. Why circumcenter? How do we prove this property? (The most important

> question, actually)

>

> And for Francois, I'm sorry but I didn't really get what you meant there.

> If possible can you provide me with the 'chowchow' that you referred to?

>

>

>

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