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Let AB and CD be 2 || line segments in space, so that ABCD is quadrilateral. ( A, B, C and are coplanar points.)
Complete the quadrilateral ABCD.
Let X be the mid point od AD and Y be mid point of BC.
Then XY is also a || line between the || lines keeping equal distance from AB and CD in the plane of ABCD.
Let L be a perpendicular plane to the plane ABCD through the line XY.
Now all the points on the plane L are equidistant from the || lines AB and CD.
So the locus of the points in space that are equidistant from two parallel lines is a plane perpendicular to the plane of the || lines. This plane passes through all the midpoints between the two || lines.
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