Parabolic polygons
  Main Theorem. Two parabols have four common points. There exists a circle tangent to the sides of the obtained parabolic quadrilateral if and only if the diagonals of this quadrilateral are orthogonal.   The proof of the Main Theorem is elementary and purely synthetic. It is based on the following lemma. Assume that a parabola is tangent to a circle at points A and B. A point P of the plane lyes on the parabola if and only if the distance from the point P to the line AB equals to the length of the tangent from P to the circle.   We present some beautiful elementary corollaries of the Main Theorem. 
F. Nilov