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Concurrent circles

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  • xpolakis
    Most likely we have seen this before, since there were here several discusions on concurrent circles. Anyway, here is it: Let ABC be a triangle and P,Q two
    Message 1 of 42 , Feb 24, 2009
      Most likely we have seen this before, since there were here
      several discusions on concurrent circles.

      Anyway, here is it:

      Let ABC be a triangle and P,Q two points.

      Denote

      A' := (reflection of BP in BQ) /\ (reflection of CP in CQ)

      B' := (reflection of P in CQ) /\ (reflection of AP in AQ)

      C' := (reflection of AP in AQ) /\ (reflection of BP in BQ)

      The Circles A'BC, B'CA, C'AB are concurrent at some point R.

      Question:
      Which are the coordinates of R in terms of the
      coordinates of P and Q.
      Special Case: P, Q = P* : Isogonal conjugate points.

      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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