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Concurrent circles

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  • Antreas
    Let ABC be a triangle and A B C the cevian triangle of I. Denote: B a, C a = the reflections of B ,C in AA , resp. A a = (perpendicular to BB from B a) /
    Message 1 of 42 , Feb 26, 2013
      Let ABC be a triangle and A'B'C' the cevian triangle of I.

      Denote:

      B'a, C'a = the reflections of B',C' in AA', resp.

      A'a = (perpendicular to BB' from B'a) /\ (perpendicular to CC' from
      C'a).

      C'b, A'b = the reflections of C',A' in BB', resp.

      B'b = (perpendicular to CC' from C'b) /\ (perpendicular to AA' from
      A'b).

      A'c, B'c = the reflections of A',B' in CC', resp.

      C'c = (perpendicular to AA' from A'c) /\ (perpendicular to BB' from
      B'c).

      Are the circumcircles of A'aB'aC'a, B'bC'bA'b, C'cA'cB'c
      concurrent?.

      Let Ia,Ib,Ic be the incenters of A'aB'aC'a, B'bC'bA'b, C'cA'cB'c,
      resp.

      Are the triangles ABC, IaIbIc perspective?


      APH
    • Antreas Hatzipolakis
      Let ABC be a triangle, P a point and A B C the pedal triangle of P. Denote: L = the Euler line of A B C A , B , C = orthogonal projections of A, B, C, on
      Message 42 of 42 , Jan 30

        Let ABC be a triangle, P a point and  A'B'C' the pedal triangle of P.

        Denote:

        L = the Euler line of A'B'C'

        A", B", C" = orthogonal projections of A, B, C, on L, resp.

        AaAbAc, BaBbBc, CaCbCc = the pedal triangles of A", B", C", wrt triangle A'B'C' resp.

        The circumcircles of AaAbAc, BaBbBc, CaCbCc are concurrent on the NPC circle of A'B'C'.
        The point of concurrence is the reflection point of L wrt the medial triangle of A'B'C'.

        Problem:

        Let ABC be a triangle and L a line.

        For which points P's the L is the Euler line of the pedal triangle of P ?

        APH
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