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22188CONCURRENT CIRCLES (Re: Three circles centered at the pedals of a point - Locus

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  • Antreas Hatzipolakis
    Apr 25, 2014
      [APH]:


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

      Denote:

      A* = the other than P intersection of the circles (B', B'P) and (C', C'P)
      B* = the other than P intersection of the circles (C', C'P) and (A', A'P)
      C* = the other than P intersection of the circles (A', A'P) and (B', B'P)



      In this configuration let Ja,Jb,Jc be the excenters of A*B*C*.

      For P = O, The triangles ABC, JaJbJc are perspective at N.

      Locus of P such that ABC, JaJbJc are perspective?

      Also, for P = O, the circumcircles of AJbJc, BJcJa, CJaJb, ABC are concurrent.
      And the circumcircles of JaBC, JbCA, JcAB too.

      If the triangle ABC is not acute, with  A > 90 d., then we take the incenter
      J of A*B*C* instead of the a-excenter Ja. ie
      The circumcircles of AJbAJc, BJcJ, CJJb, ABC are concurrent.

      Antreas





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