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Locus of midpoints in space

Source: Czech and Slovak Olympiad 1953, National Round, Problem 4

February 15, 2022
3D geometryLocus

Problem Statement

Consider skew lines a,ba,b and a plane ρ\rho that intersect both of the lines (but does not contain any of them). Choose such points X∈a,Y∈bX\in a,Y\in b that XY∥ρ.XY\parallel\rho. Find the locus of midpoints MM of all segments XY,XY, when XX moves along line aa.
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