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 1 of 5: Write the three midpoint vectors
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In plain words

A midpoint is the average of its two endpoints.

L=A+A′2,M=B+B′2,N=C+C′2L=\frac{A+A'}{2},\qquad M=\frac{B+B'}{2},\qquad N=\frac{C+C'}{2}
Detailed analysis

Use vector coordinates in space. Since L,M,NL,M,N are the respective midpoints, their position vectors are exactly the three displayed averages.