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]