Parabola focus and directrix

Moira Chas

Drag the point D to see points in the parabola highlighted.

Recall that a parabola is the set of points equidistant from a line (directrix) and point (focus) We are constructing a parabola with focus C and as directrix the red line. We chose a point D in the directrix and we construct the perpendicular bisector of C and D. This is the lilac line. We construct the perpendicular to the directrix though D. We mark the point E, which is the intersection of the perpendicular bisector and the perpendicular to the directrix through D.All points E obtained in such a way (by varying D on the directrix) form a parabola.Let's prove that the set set of point E obtained with the above construction form a parabola.Consider a point E in the parabola.  Construct a perpendicular to the directrix through E. D is the intersection point of this perpendicular and the directrix. The the length of ED is the distance between E and the directrix.