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 3 of 5: Recognize two centroids
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G=12(A+B+C3+A′+B′+C′3)G=\frac12\left(\frac{A+B+C}{3}+\frac{A'+B'+C'}{3}\right)
Detailed analysis

Let T=(A+B+C)/3T=(A+B+C)/3 be the centroid of ABCABC and T′=(A′+B′+C′)/3T'=(A'+B'+C')/3 the centroid of A′B′C′A'B'C'. Then G=(T+T′)/2G=(T+T')/2, so GG is the midpoint of TT′TT'.