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 5 of 5: Identify the locus plane
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L(G)={(T+T′)/2:T′∈ϵ}\mathcal L(G)=\{(T+T')/2:T'\in\epsilon\}
Detailed analysis

The homothety with center TT and ratio 1/21/2 maps the plane ϵ\epsilon to a plane parallel to ϵ\epsilon, halfway between TT and ϵ\epsilon. Since T′T' ranges over all of ϵ\epsilon, this image is exactly the locus of GG.