The focus-directrix construction of a parabola

You can see the steps in the Construction Protocol view. Here they are in English, too.[br]Construct a point A and a line BC that doesn't contain A.[br]Construct a point D on BC, not B or C.[br]Construct the segment AD, then the perpendicular bisector of AD.[br]Construct the perpendicular to BC at D.[br]Intersect the last two lines (perpendicular bisector and perpendicular) to get E.[br]Now form the [i]locus[/i] (that is, the trace) of point E as its controller, D, moves along BC:[br] Choose the Locus tool, at the bottom of the perpendicular set of tools[br] Click on the point to trace, E, then on its controlling point, D.[br]A curve appears. As point D moves along BC, point E traces out the curve.[br]If you change A, B, or C, the shape of the curve changes.[br][br]This is a parabola (by definition). It is the set (locus) of points that are equidistant from a point (the focus) and a line (the directrix). Why?[br][br]Hints: The perpendicular bisector of a segment is the set of points equidistant from the endpoints of the segment.[br]How do you measure the distance of a point (E) to a line (BC)?[br]

Information: The focus-directrix construction of a parabola