I need a solution for three dimensions. This is not an assignment and I've been through google without success. I'm sure I have a book laying around with the answer, but I can't seem to find it.

The point of intersection of the perpendicular bisector and the original line is exactly halfway along the first line. If everything has been constructed correctly, the two lines will always meet ...

Learn how to find equations of parallel lines, work out the gradient of perpendicular lines, and prove that given lines are parallel or perpendicular.