Points of the circumcircle with Simson line through a given point P

1) Draw the rectangular circumhyperbola through P.
2) Translate this hyperbola following the vector HP (H being the orthocenter).
3) The translated hyperbola intersects the circumcircle at four points, one of them also in the first hyperbola. The other three points are have Simson line through P.
If the circumcircle and the hyperbola have only two intersection points, there is only one point having Simson line through P.