Quadrilateral Inscribed in Orthogonal Parabolas


Construct two parabolas so that the directrix of one is perpendicular to the directrix of the other. The parabolas can intersect in at most 4 points. Explore how your construction could find a case with 4 intersection points. Create the quadrilateral determined by those 4 points.

Show that the quadrilateral is CYCLIC. (A cyclic quadrilateral is one where the four vertices lie on a circle.)

a. Do this investigation using algebra and graphs that are equation driven. Implement with a graphing program

b. Do this investigation using geometric constructions and synthetic geometry reasoning. Implement with Geometer's Sketchpad.

c. Constrast the two approaches. This problem as formulated geometrically by the ancients and Omar Khyyam used it in building his algebraic solutions to cubic equations. Clearly, it was a geometry problem for which later developments in coordinate geometry allowed the use of algebraic methods.