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 2 of 5: Compute the centroid of LMNLMN
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G=L+M+N3=A+B+C+A′+B′+C′6G=\frac{L+M+N}{3}=\frac{A+B+C+A'+B'+C'}{6}
Detailed analysis

The centroid is the average of the three vertices. Substituting the midpoint formulas and collecting terms gives the second expression.