nine-point circle and parabolic segment
  Given the focus F and the directrix d of a parabola. Consider a parabolic segment SE and the tangents at its endpoints intersecting at C:
- point M is the middle of SE,
- line CCd is perpendicular to SE,
- point Cd is intersection of directix and CCd.

Then MCd is the diameter of  nine-point circle of triangle SCE.
Second intersection of nine-point circle and directrissa (at point Md) is projection of point M on the directrissa.
   
Ivan Dembicki