Re: [EMHL] Re: Median line
- Le 08/03/2011 19:09, jpehrmfr a écrit :
>> Is the line L, minimizing the sum ofYes, and is the line maximizing the sum the other
>> 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
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.
- The following paper has been published in Forum Geometricorum. It can be viewed at
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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