MathLabs

Problem 6

Consider a plane ϵ\epsilon and three non-collinear points A,B,CA,B,C on the same side of it; the plane ABCABC is not parallel to ϵ\epsilon. In ϵ\epsilon choose arbitrary points A′,B′,C′A',B',C'. Let L,M,NL,M,N be the midpoints of AA′,BB′,CC′AA',BB',CC', and let GG be the centroid of triangle LMNLMN. Find the locus of GG as A′,B′,C′A',B',C' range independently over ϵ\epsilon, excluding degenerate LMNLMN.
Step 4 of 5: Show that the second centroid fills the plane
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T′∈ϵ arbitrarilyT'\in\epsilon\text{ arbitrarily}
Detailed analysis

The centroid of three independently chosen points in an affine plane can be any point of that plane: for a desired T′T', choose for example A′=B′=C′=T′A'=B'=C'=T'. Nondegenerate choices can be made arbitrarily close while retaining the same centroid, so excluding degenerate LMNLMN does not remove any locus point.