Orthogonal Parabolas

Two parabolas are orthogonal if the directrix of one is perpendicular to the other. Two orthogonal parabolas could intersect in 0, 1, 2, 3, or 4 points:

These five illustrations are GSP constructions based on the definition of a parabola as the locus of points equidistant from a line called the directrix and a point called the focus.

We could have also illustrated these examples with parabolas generated by equations:

(here we have omitted the focal points and the directrices)

PROBLEM:

For the case where orthogonal parabolas intersect in four points, show that the quadrilateral determined by those four points is cyclic. That is, there is a circle that lies on the four vertices of the quadrilateral.

1. Show this result via algebra

2. Show the result using a geometric argument.