Take a line and a point F not on the line. From a point on the line create a segment to F. Create the perpendicular bisector of this segment.
The envelope of these perpendicular bisectors using every point on the line is a parabola.
If we create perpendiculars that are some ratio, r, of the distance from the line to the point besides 1/2, what shape will the envelope become?
Part 2.
Take a line and a point F not on the line. Construct the set of all points equidistant from the point and line.
This set of points is a parabola.
If we find all the points that are d times the distance from the point as they are to the line, what shape will this set of points become?
This page indicates both constructions with a single interactive.
