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Suppose that in a Hilbert plane, lines m and n are parallel. Let P be a...

Suppose that in a Hilbert plane, lines m and n are parallel. Let P be a point on m, and Q the point on n so that line P Q is perpendicular to n. Is there another point R on m so that the perpendicular line RS to n through R (where S is incident with n) so that segment P Q is congruent to RS? Under what conditions? (Note that you need to provide some further hypotheses to make a clear statement about the existence or non-existence of such additional points.)

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