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22055Pedal Triangle - Parallels to Sidelines - Locus

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  • Antreas Hatzipolakis
    Mar 15, 2014
      Let ABC be a triangle, P a point and PaPbPc the pedal
      triangle of P.

      Ab = The orthogonal projection of Pa on PPb
      Ac = The orthogonal projection of Pa on PPc

      A2 = (Parallel to BC through Ab) /\ AC
      A3 = (Parallel to BC through Ac) /\ AB

      M1 = The midpoint of A2A3

      Similarly (Cyclically) M2, M3

      Which is the locus of P such that

      1. Perpendicular bisectors of A2A3, B3B1, C1C2 are concurrent?

      2. ABC, M1M2M3 are orthologic?

      3. M1M2M3, Orthic Triangle of ABC are perspective?

      4. ABC, M1M23 are perspective?

      The simplest case should be for P = O, since Ab = Cb, Bc = Ac
      and Ba = Ca.