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Re: Radical Axes and Loci

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  • jpehrmfr
    Dear Antreas and other Hyacinthists [Antreas] ... axes ... concurrent. ... Some easy remarks : - (O) is the circumcircle, (A) is the circle (A,r1),.. - UVW is
    Message 1 of 8 , Mar 30 12:53 AM
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      Dear Antreas and other Hyacinthists
      [Antreas]
      > For any three circles (A,r1), (B,r2), (C,r3) centered on A,B,C the radical
      axes
      > of (A,r1), (B,r2)), ((B,r2), (C,r3)), ((C,r3),(A,r1)) are obviously
      concurrent.
      >
      > So we can always ask the same questions for loci.
      >
      > But the problem is to define interesting radii of the circles!!

      Some easy remarks :
      - (O) is the circumcircle, (A) is the circle (A,r1),..
      - UVW is the triangle bounded by the radical axis of the circles
      (O)&(A),(O)&(B),(O)&(C)
      - O' is the circumcenter of UVW
      - D is the radical center of (A),(B),(C)
      1) DO' is parallel to the Euler line of ABC
      Explanation : the lines VW,WU,UV are perpendicular respectively to OA,OB,OC;
      thus UVW and the orthic triangle A'B'C' are homothetic and the homothecy h maps
      the 9P-center N to O'.
      As U is the radical center of (O),(B),(C), DU&BC are perpendicular; so do DV&CA, DW&AB. Hence D = h(H) and the result
      We can notice too that D is the incenter (or an excenter if ABC is obtusangle)
      of UVW and that O & D are the orthologic centers of ABC & UVW
      2) The locus of the points M for which
      (b^2-c^2)AM^2+(c^2-a^2)BM^2+(a^2-b^2)CM^2=0 is a line through O,G,H; so it is
      the Euler line of ABC
      As AD^2-r1^2 = BD^2-r2^2 = CD^2-r3^2, we have
      (b^2-c^2)AD^2+(c^2-a^2)BD^2+(a^2-b^2)CD^2 =
      (b^2-c^2)r1^2+(c^2-a^2)r2^2+(a^2-b^2)r3^2
      It follows from 1) and 2) that we have 3 equivalent conditions :
      a) O' lies on the Euler line of ABC
      b) D lies on the Euler line of ABC
      c) (b^2-c^2)r1^2+(c^2-a^2)r2^2+(a^2-b^2)r3^2 = 0
      So, we can consider that c) is an interesting condition for the radii
      Friendly. Jean-Pierre
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