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22476Isogonal Conjugates - NPC - Radical Axis

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  • Antreas Hatzipolakis
    Jun 29, 2014
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      Let ABC be a triangle, P,P* two isogonal conugate points
      and PaPbPc, P*aP*bP*c the pedal triangles of P, P*, resp.

      Denote:

      Ab = the orthogonal projection of Pa on AC
      A2 = the reflection of A in Ab
      (= the other than A intersection of the circle (Pa, PaA) and AC)

      Ac = the orthogonal projection of Pa on AB
      A3 = the reflection of A in Ac
      (= the other than A intersection of the circle (Pa, PaA) and AB)

      (Na) = the NPC of AA2A3

      A*b = the orthogonal projection of P*a on AC
      A*2 = the reflection of A in A*b
      (= the other than A intersection of the circle (P*a, P*aA) and AC)

      A*c = the orthogonal projection of P*a on AB
      A*3 = the reflection of A in A*c
      (= the other than A intersection of the circle (P*a, P*aA) and AB)

      (N*a) = the NPC of AA*2A*3

      Ra = the radical axis of (Na) and (N*b)

      Similarly Rb and Rc.

      The Ra,Rb,Rc are concurrent.
      The parallels to Ra,Rb,Rc through A,B,C, resp. concur at a fixed point
      [X(74)]

      Antreas