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21965Forum Geometricorum

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  • forumgeom forumgeom
    Apr 16, 2013
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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 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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