Parabola 9 Support Begin with the given points P1, P2, and P3, and the line diameter. Through point P1 construct the line k, perpendicular to the given diameter. Construct circles centered on P2 and P3 (in green), tangent to k. The directrix must be parallel to k, either above all three points on curve or below all three. The distances from points P2 and P3 to k must be how much nearer to, or further from, the directrix, compared with the distance of P1. The system of point P1, circle P2, and circle P3 has a radical center, C, center of the circle (in red) through P1 and orthogonal to circles P2 and P3. Point C actually lies on the axis of the parabola (to be shown). Let circle F (in blue) be the inversion of line k with respect to circle C. Point P1 and circles P2 and P3 are invariant in that transformation, so circle F passes through P1 and is tangent to circles P2 and P3, just as k. Circle F also passes through C, and FC is a diameter of the parabola. Let r2 and r3 be the radii of circles P2 and P3 (the green circles above). Compare the distances of the three given points to point F.   Construct line m parallel to k and offset from it by distance FP1. The distances above are also the displacements of the three points from this line, making all three points equidistant from point F and line m, as illustrated below. Thus, F and m are the focus and directrix. Last update: May 12, 2026 ... Paul Kunkel whistling@whistleralley.com |