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22162Re: Envelopes

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  • Antreas Hatzipolakis
    Apr 20, 2014
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      On Sun, Apr 20, 2014 at 8:13 PM, Antreas Hatzipolakis <anopolis72@...> wrote:


      The problem is:
      Let ABC be a triangle and L1,L2,L3 three parallel lines through A,B,C, resp. and let A'B'C' be the triangle bounded by the reflections of L1,L2,L3, in BC,CA,AB, resp.
      As the lines L1,L2,L3 move around A,B,C, being parallel, which is the envelope of the Euler line of A'B'C'?
      Furthermore, which is the envelope of the circumcircle of A'B'C'?

      APH


      The problem was posed by the friend Dao Thanh Oai, in my wording:
      Let ABC be a triangle and L1,L2,L3 three parallel lines through A,B,C, resp. and let A'B'C' be the triangle bounded by the reflections of L1,L2,L3, in BC,CA,AB, resp.
      As the lines L1,L2,L3 move around A,B,C, the Euler line of the triangle
      A1B1C1 passes through a fixed point.

      At first glance I thought that if it was true, then the fixed point should be
      the Parry Point X(399), the point the reflections in BC,CA,AB of the parallels
      to Euler line through A,B,C, resp. concur at.

      But I was not right!!! If the lines L1,L2,L3 are parallels to Euler line,
      then the triangle A1B1C1 is degenerated in Parry point, and its Euler line
      is ANY line passing through Parry point!

      The fixed point has barycentrics (computed by Francisco Javier García Capitán):

      The point is (f(a,b,c):f(b,c,a):f(c,a,b)) where f(a,b,c) is
      a^22 - 8 a^20 b^2 + 28 a^18 b^4 - 56 a^16 b^6 + 70 a^14 b^8 -
      56 a^12 b^10 + 28 a^10 b^12 - 8 a^8 b^14 + a^6 b^16 - 8 a^20 c^2 +
      42 a^18 b^2 c^2 - 92 a^16 b^4 c^2 + 106 a^14 b^6 c^2 -
      62 a^12 b^8 c^2 + 7 a^10 b^10 c^2 + 13 a^8 b^12 c^2 -
      8 a^6 b^14 c^2 + 4 a^4 b^16 c^2 - 3 a^2 b^18 c^2 + b^20 c^2 +
      28 a^18 c^4 - 92 a^16 b^2 c^4 + 113 a^14 b^4 c^4 - 62 a^12 b^6 c^4 +
      17 a^10 b^8 c^4 - 9 a^8 b^10 c^4 + 5 a^6 b^12 c^4 - 6 a^4 b^14 c^4 +
      13 a^2 b^16 c^4 - 7 b^18 c^4 - 56 a^16 c^6 + 106 a^14 b^2 c^6 -
      62 a^12 b^4 c^6 + 4 a^10 b^6 c^6 + 4 a^8 b^8 c^6 + 8 a^6 b^10 c^6 -
      6 a^4 b^12 c^6 - 18 a^2 b^14 c^6 + 20 b^16 c^6 + 70 a^14 c^8 -
      62 a^12 b^2 c^8 + 17 a^10 b^4 c^8 + 4 a^8 b^6 c^8 - 12 a^6 b^8 c^8 +
      8 a^4 b^10 c^8 + 3 a^2 b^12 c^8 - 28 b^14 c^8 - 56 a^12 c^10 +
      7 a^10 b^2 c^10 - 9 a^8 b^4 c^10 + 8 a^6 b^6 c^10 + 8 a^4 b^8 c^10 +
      10 a^2 b^10 c^10 + 14 b^12 c^10 + 28 a^10 c^12 + 13 a^8 b^2 c^12 +
      5 a^6 b^4 c^12 - 6 a^4 b^6 c^12 + 3 a^2 b^8 c^12 + 14 b^10 c^12 -
      8 a^8 c^14 - 8 a^6 b^2 c^14 - 6 a^4 b^4 c^14 - 18 a^2 b^6 c^14 -
      28 b^8 c^14 + a^6 c^16 + 4 a^4 b^2 c^16 + 13 a^2 b^4 c^16 +
      20 b^6 c^16 - 3 a^2 b^2 c^18 - 7 b^4 c^18 + b^2 c^20

      https://www.facebook.com/photo.php?fbid=1498558183700588&set=gm.437719496364497&type=1&theater


      How about the envelope of the circumcircle of the triangle A1B1C1 ?

      Antreas
       
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