GEOMETRIC CONSTRUCTION OF A PARABOLA

Emat 6680 - Assignment 6

by Mary H. Bruce

In the high school curriculum, a student is often introduced to a parabola from an algebraic perspective when learning quadratic equations.  In this assignment the geometric construction of a parabola will be examined.  A parabola is the set of all points that are equidistant from a fixed point (the focus) and a line (the directrix).

Given a fixed point F for the focus and segment AB for the directrix, construct a perpendicular line through a random point E on segment AB as shown:

Next form segment FE and construct the perpendicular bisector of this segment (line m):

Find the point of intersection of line m and the green line (call it point P).  Then connect segment PF and segment PE, the congruent sides of the isosceles triangle formed, thus constructing a point P equidistant from the focus and the directrix:

As point E moves along segment AB, the trace of the equidistant point P is generated.  Click here to view animation and trace of point P and resulting parabola.  Click here to view trace of the tangent line at point P and resulting parabola.

In the final sketch, GSP is used to generate the locus of points equidistant from a fixed point (focus) and the directrix line: