Darren C. Ong, On a theorem of intersecting conics,
Forum Geometricorum, 11 (2011) 95--107.
Abstract. Given two conics over an infinite field that intersect at the origin, a line through the origin will, in general intersect both conic sections once more each, at points C and D. As the line varies we find that the midpoint of C and D traces out a curve, which is typically a quartic. Intuitively, this locus is the ``average" of the two conics from the perspective of an observer at the origin. We give necessary and sufficient conditions for this locus to be a point, line, line minus a point, or a conic itself.
[ps file] [pdf]