Conic Sections and Regular Pentagons Inscribed in Circles

We present some results on conic sections that intersect a given circle at four points which, together with a specified point on the circle, form the vertices of a regular pentagon inscribed in the circle. For a circle C and a point A on C, there exists a unique parabola P such that the four intersections of C and P, together with A, form the vertices of a regular pentagon inscribed in C. For two circles of the same radius tangent at a point A, there exists a unique ellipse E such that the intersections of the circles and E, together with A, form the vertices of a regular pentagon inscribed in each circle. Also, for a circle C and a point A on C, there exists a unique conic section H, which turns out to be a hyperbola, going through the center of the circle such that the four intersections of C and H, together with A, form the vertices of a regular pentagon inscribed in C. The results are stated in the languages of elementary geometry and analytic geometry in the Euclidean plane.