Witch of Agnesi

Witch of Agnesi is the locus (trace) of point F while point D is moving on a circle. It is an algebraic (polynomial) curve with a degree of 3. The curve contains just one part, but in the figure there is an extra component (a line) because of some technical reasons. Thus the equation of the curve in the figure has a degree of 4.
  1. Move point D on the circle to get different tracer points for F. Can you describe the Euclidean steps how point F is constructed?
  2. Drag point C to change the position of line b.
  3. Drag point B out of the y-axis to get rotated versions of the curve. Check if the equation has still a degree of 4.