Points being the ninecenter of their pedal triangle

1) Let D be the foot of the A-altitude. Take Db, Dc on BC such that BDb/BD = CDc/CD = -2.
2) The parallel from B to ADb cuts AC at E.
3) The parallel from C to ADc cuts AB at F.
4) The bisectors of BAC cut BC at K, L.
5) Let Ha be the hyperbola {AKLEF}. Define analogously Hb, Hc.
6) The three hyperbolas intersect at the four possible positions of the requested point.