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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