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22157A quadrangle of quadrangle

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  • Antreas Hatzipolakis
    Apr 19 7:03 PM
      I quote my message #21938 to Hyacinthos:

      Let ABC be a triangle, P, P* two isogonal conjugate points.

      Denote:

      Ra = radical axis of (NPC_PBC), (NPC_P*BC)

      Rb = radical axis of (NPC_PCA), (NPC_P*CA)

      Rc = radical axis of (NPC_PAB), (NPC_P*AB)

      Which is the locus of P such that Ra,Rb,Rc are concurrent?
      The entire plane?
      [NPC_PBC means the Nine Point Circle of the triangle PBC]

      https://groups.yahoo.com/neo/groups/Hyacinthos/conversations/messages/21938

      As it was proved the locus is indeed the whole plane.

      We can apply this to a quadrangle ABCD to get another quadrangle.

      Let D' be the point of concurrence of the above radical axes in triangle ABC and point D
      and similarly C', B',A'.

      Which properties has the quadrangle A'B'C'D'?

      Are the lines AA', BB', CC', DD' concurrent?

      Antreas