Construct a Segment with a Given Midpoint and Endpoints on Two Given Lines

Given two lines and and a point , construct a segment such that , and is its midpoint.
Construction
Choose any point on such that is not perpendicular to .
Draw a circle with the center and the radius . Let a point be the second intersection of and .
Let and be symmetric to and with respect to . Then the quadrilateral is a rectangle.
Let be the intersection of and the line . Then the point symmetric to with respect to is on and is the midpoint of .

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