## Re: [EMHL] Re: Median line

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• Dear Jean-Pierre Very nice remark about the minimum property of the focal axis of the Steiner ellipses. Have we some generalisation with the minimum of the
Message 1 of 476 , Mar 10, 2011
Dear Jean-Pierre
Very nice remark about the minimum property of the focal axis of the Steiner
ellipses.
Have we some generalisation with the minimum of the function defined on the
space of affine lines:
L --> p.d(A, L)� + q.d(B, L)� + r.d(C, L)� where the sum p+q+r is not equal
to 0.
Obviously, the minimum line is on the point with barycentrics (p, q, r).
It would be nice if the minimum line is the focal axis of the inconic (or
circcumconic) with center (p, q, r)?
Friendly
Francois

On Thu, Mar 10, 2011 at 1:32 PM, Etienne Rousee <etienne@...> wrote:

>
>
> Le 08/03/2011 19:09, jpehrmfr a �crit :
>
> >> Is the line L, minimizing the sum of
> >> the squares of the distance from A,B,C
> >> to L, always a median of ABC ?
> >
> > No, this line is the focal axis of the Steiner ellipses
>
> Yes, and is the line maximizing the sum the other
> axis of the Steiner ellipse ?
> I was mistaken because I experimented whis the sum
> of the distances and forgot the squares.
> In this case, it seems near a median.
>
> --
>
> Etienne
>
>

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• The following paper has been published in Forum Geometricorum. It can be viewed at http://forumgeom.fau.edu/FG2013volume13/FG201309ndex.html The editors Forum
Message 476 of 476 , Apr 16, 2013
The following paper has been published in Forum Geometricorum. It can be viewed at

http://forumgeom.fau.edu/FG2013volume13/FG201309ndex.html

The editors
Forum Geometricorum

Paul Yiu, On the conic through the intercepts of the three lines through the centroid and the intercepts of a given line,
Forum Geometricorum, 13 (2013) 87--102.

Abstract. Let L be a line intersecting the sidelines of triangle ABC at X, Y, Z respectively. The lines joining these intercepts to the centroid give rise to six more intercepts on the sidelines which lie on a conic Q(L,G). We show that this conic (i) degenerates in a pair of lines if L is tangent to the Steiner inellipse, (ii) is a parabola if L is tangent to the ellipse containing the trisection points of the sides, (iii) is a rectangular hyperbola if L is tangent to a circle C_G with center G. We give a ruler and compass construction of the circle C_G. Finally, we also construct the two lines each with the property that the conic Q(L,G) is a circle.

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