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22495Re: Reflections in altitudes - Locus

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  • Antreas Hatzipolakis
    Jul 3, 2014






      [APH]

      Let ABC be a triangle and P a point.

      Denote:

      P1 = the reflection of P in AH
      Pa = the reflection of P1 in BC
      Ppa = the reflection of Pa in PA

      Ma = the midpoint of PPpa

      Similarly Mb, Mc

      Which is the locus of P such that ABC, MaMbMc are perspective?

      If P = I, then they are, with perspector the circumcenter O.
      Also for P = N the triangles are perspective.
      [the congruent circles (Nna) and (N) are tangent,
      so Ma is the tangency point. Where:
      (N1) = the reflection of the NPC (N) in AH
      (Na) = the reflection of (N1) in BC
      (Nna) = the reflection of (Na) in NA)]


      In fact Ppa, Ppb, Ppc  are the reflections of the feet of
      altitudes in PA, PB, PC, resp.

      So the proplem is:

      Which is the locus of P such that the reflections of
      the altitudes AA',BB',CC' in AP, BP, CP, resp. are concurrent?
      (I think it was discussed in hyacinthos)

      APH

       
      For P = N:

      The reflections of AH,BH,CH in AN,BN,CN are concurrent
      APH
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