Re: Envelope problem
- Dear Antreas
> Le ABC be a triangle, (c) a curve on the planeThe comnon points between the NPC and the pedal circle of P are the centers of the rectangular circumhyperbolae through P and through P* (isogonal conjugate of P)
> of ABC, and P a point on (c).
> General Problem:
> Which is the envelope of the radical axis of the NPC of ABC
> and the pedal circle of P, as P moves on (c)?
> If (c) is the McCay cubic, then the two circles are tangent,
> and therefore the envelope is the NPC.
> How about simple (c)'s?
> For example, (c) := the Euler line of ABC.
Now, if P moves on a line L going through O, the rectangular circumhyperbola going through P* is the isogonal conjugate of L (and doesn't depend on the position of P upon L); hence the radical axis goes through a fixed point of the NPC : the center of the isogonal conjugate of L
In the case of the Euler line, the fixed point is the center of the Jerabek hyperbola