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FEUERBACH POINT

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  • Antreas Hatzipolakis
    http://anthrakitis.blogspot.gr/2013/02/feuerbach-point.html Synthetic proofs? Let ABC be a triangle and A B C , A B C the cevian, pedal triangles of I, resp.
    Message 1 of 1 , Feb 25, 2013
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      http://anthrakitis.blogspot.gr/2013/02/feuerbach-point.html

      Synthetic proofs?


      Let ABC be a triangle and A'B'C', A"B"C" the cevian, pedal triangles of I, resp.

      Denote:

      Ab, Ac = the reflections of A' in BB', CC'

      Bc, Ba = the reflections of B' in CC', AA'

      Ca, Cb = the reflections of C' in AA', BB'

      Ea, Eb, Ec = the Euler Lines of A'AbAc, B'BcBa, C'CaCb, resp. (concurrent at I)

      1. Feuerbach Point Antipode.

      Denote:

      A2,A3 = the orthogonal projections of A on Eb, Ec, resp.

      B3,B1 = the orthogonal projections of B on Ec, Ea, resp.


      C1,C2 = the orthogonal projections of C on Ea, Eb, resp.

      The Euler lines of AA2A3, BB3B1, CC1C2 are concurrent at the antipode
      of the Feuerbach point.

      Note:

      Denote:

      12, 13 = the orthogonal projections of A on BB', CC', resp.

      23, 21 = the orthogonal projections of B on CC', AA', resp.

      31, 32 = the orthogonal projections of C on AA', BB', resp.

      The Euler lines of A1213, B2321, C3132 are concurrent at Feuerbach point

      (APH, Hyacinthos)

      2. Feuerbach Point Antipode.

      Denote:


      La = the parallel to Ea through A"

      Lb = the parallel to Eb through B"

      Lc = the parallel to Ec through C"

      The La,Lb,Lc are concurrent at the antipode of the Feuerbach point.

      Antreas
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