1961 IMO Problems/Problem 6
Problem
Consider a plane and three non-collinear points
on the same side of
; suppose the plane determined by these three points is not parallel to
. In plane
take three arbitrary points
. Let
be the midpoints of segments
; Let
be the centroid of the triangle
. (We will not consider positions of the points
such that the points
do not form a triangle.) What is the locus of point
as
range independently over the plane
?
Solution
We will consider the various points in terms of their coordinates in space. We have . Since the centroid of a triangle is the average of the triangle's vertices, we have
. It is clear now that
is midpoint of the line segment connecting the centroid of
and the centroid of
. It is obvious that the centroid of
can be any point on plane
. Thus, the locus of
is the plane parallel to
and halfway between the centroid of
and
.
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