High Quality Content by WIKIPEDIA articles! In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... of one and the same line are isometrically mapped to points P', Q', R'... of another line in the same plane, then the midpoints of the segments PP', QQ', RR'... also lie on one and the same line. The proof is easy if one assumes the classification of plane isometries. If the given isometry is odd, in which case it is necessarily either a reflection in a line or a glide-reflection (the product of three reflections in a line and two perpendiculars to it), then the statement is true of any points in the plane whatsoever: the midpoint of PP' lies upon the axis of the (glide-)reflection for any P. If the isometry is even, compose it with reflection in line PQR to obtain an odd isometry with the same effect on P, Q, R... and apply the previous remark.

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