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14774Re: Another perspector?

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  • garciacapitan
    Jan 23, 2007
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      > If we consider the three small extangent circles (O_1),(O_2),(O_3)
      and
      > NPC of ABC, then the radicals axis to the three circles (O_1),(O_2),
      > (O_3) with NPC performed a triangle QST, hence QST and ABC are
      > perspective at P'.
      > Is P'in ETC?


      Dear Juan Carlos and all

      Related to this problem I can give a construction that uses inversion
      and the coordinates of your point P', not in ETC.

      The construction can be found as a solution of problem 301 in Ricardo
      Barroso's online magazine

      http://www.aloj.us.es/rbarroso/trianguloscabri/sol/sol301garcap/

      (server is slow, so you may need try it several times)

      I gave there a formula for the radius of (O_1)

      A rough translation of the Spanish text follows.

      -------------------------------------------------------------------
      Let K, L be the tangency points of (Ic),(Ib) with line BC.
      We consider an inversion with center K and radius KL (or power KL^2).

      The triangle KLIb is rectangle at L, so the circle (Ib) is orthogonal
      to the inversion circle and thus invarible by the inversion. The circle
      (Ic) pass trough the inversion center, so circle (Ic) inverts in a
      line, namely, if KM is a diameter of (Ic), then circle (Ic) inverts in
      the parallel to BC through M', the inverse of M.

      The circle we want is the inverse of the circle (T) tangent to this
      parallel, the line BC and the circle (Ib).

      Then the center T must be in the median parallel to two lines. Let this
      median parallel intersect KM at N. Then T is KN + rb away from Ib, and
      this let us get the center T.
      -------------------------------------------------------------------

      To end this long post I give the coordinates of P', not in ETC.

      Search Number is 1.367790355

      We name 2 sqrt((s-b)(s-c)) as RA, and RB, RC the same ciclically.

      {(a + b - c)*(a - b + c)*(b + c)*(2*a^4 + a^2*b^2 + b^4 + 2*a^2*b*c -
      3*a^2*c^2 - 2*b^2*c^2 + c^4 - 2*a^3*RA - 2*a*b^2*RA +
      2*a*c^2*RA)*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 - 2*a^5*c -
      3*a^4*b*c + 2*a^3*b^2*c + a^2*b^3*c - 2*a*b^4*c - 2*a^3*b*c^2 -
      2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 + 6*a^2*b*c^3 +
      2*a*b^2*c^3 - b^3*c^3 + a*b*c^4 - 3*b^2*c^4 - 2*a*c^5 -
      3*b*c^5 - a^5*RB - a^4*b*RB + a^3*b^2*RB + a^2*b^3*RB - a^4*c*RB -
      4*a^3*b*c*RB - a^2*b^2*c*RB + 2*a*b^3*c*RB +
      2*a^3*c^2*RB + 2*a^2*b*c^2*RB - a*b^2*c^2*RB + b^3*c^2*RB +
      2*a^2*c^3*RB + 4*a*b*c^3*RB + b^2*c^3*RB - a*c^4*RB - b*c^4*RB -
      c^5*RB)*(2*a^5*b - 4*a^3*b^3 + 2*a*b^5 + 3*a^5*c - a^4*b*c -
      6*a^3*b^2*c + 2*a^2*b^3*c + 3*a*b^4*c - b^5*c + 3*a^4*c^2 -
      2*a^3*b*c^2 + 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^3*c^3 -
      a^2*b*c^3 - a*b^2*c^3 + b^3*c^3 + a^2*c^4 + 2*a*b*c^4 +
      b^2*c^4 + a^5*RC + a^4*b*RC - 2*a^3*b^2*RC - 2*a^2*b^3*RC +
      a*b^4*RC + b^5*RC + a^4*c*RC - 4*a^3*b*c*RC - 2*a^2*b^2*c*RC +
      4*a*b^3*c*RC + b^4*c*RC - a^3*c^2*RC + a^2*b*c^2*RC + a*b^2*c^2*RC
      - b^3*c^2*RC - a^2*c^3*RC - 2*a*b*c^3*RC - b^2*c^3*RC),
      (a - b - c)*(a + b - c)*(a + c)*(a^4*b^2 + a^3*b^3 + 3*a^2*b^4 +
      3*a*b^5 + 2*a^4*b*c - a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c +
      2*b^5*c + a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - 6*a*b^3*c^2 +
      a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 +
      3*a*b*c^4 - a*c^5 + 2*b*c^5 - a^3*b^2*RA - a^2*b^3*RA + a*b^4*RA +
      b^5*RA - 2*a^3*b*c*RA + a^2*b^2*c*RA - 4*a*b^3*c*RA +
      b^4*c*RA - a^3*c^2*RA + a^2*b*c^2*RA - 2*a*b^2*c^2*RA -
      2*b^3*c^2*RA - a^2*c^3*RA + 4*a*b*c^3*RA - 2*b^2*c^3*RA + a*c^4*RA +
      b*c^4*RA + c^5*RA)*(a^4 - 3*a^2*b^2 + 2*b^4 + 2*a*b^2*c - 2*a^2*c^2
      + b^2*c^2 + c^4 + 2*a^2*b*RB - 2*b^3*RB - 2*b*c^2*RB)*
      (2*a^5*b - 4*a^3*b^3 + 2*a*b^5 + 3*a^5*c - a^4*b*c - 6*a^3*b^2*c +
      2*a^2*b^3*c + 3*a*b^4*c - b^5*c + 3*a^4*c^2 - 2*a^3*b*c^2 +
      2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^3*c^3 - a^2*b*c^3 -
      a*b^2*c^3 + b^3*c^3 + a^2*c^4 + 2*a*b*c^4 + b^2*c^4 + a^5*RC +
      a^4*b*RC - 2*a^3*b^2*RC - 2*a^2*b^3*RC + a*b^4*RC + b^5*RC +
      a^4*c*RC - 4*a^3*b*c*RC - 2*a^2*b^2*c*RC + 4*a*b^3*c*RC +
      b^4*c*RC - a^3*c^2*RC + a^2*b*c^2*RC + a*b^2*c^2*RC - b^3*c^2*RC -
      a^2*c^3*RC - 2*a*b*c^3*RC - b^2*c^3*RC),
      -((a + b)*(a - b - c)*(a - b + c)*(a^4*b^2 + a^3*b^3 + 3*a^2*b^4 +
      3*a*b^5 + 2*a^4*b*c - a^3*b^2*c - 2*a^2*b^3*c - a*b^4*c +
      2*b^5*c + a^4*c^2 - a^3*b*c^2 + 2*a^2*b^2*c^2 - 6*a*b^3*c^2 +
      a^3*c^3 - 2*a^2*b*c^3 + 2*a*b^2*c^3 - 4*b^3*c^3 - a^2*c^4 +
      3*a*b*c^4 - a*c^5 + 2*b*c^5 - a^3*b^2*RA - a^2*b^3*RA + a*b^4*RA +
      b^5*RA - 2*a^3*b*c*RA + a^2*b^2*c*RA - 4*a*b^3*c*RA +
      b^4*c*RA - a^3*c^2*RA + a^2*b*c^2*RA - 2*a*b^2*c^2*RA -
      2*b^3*c^2*RA - a^2*c^3*RA + 4*a*b*c^3*RA - 2*b^2*c^3*RA + a*c^4*RA +
      b*c^4*RA + c^5*RA)*(a^5*b + a^4*b^2 - a^3*b^3 - a^2*b^4 - 2*a^5*c
      - 3*a^4*b*c + 2*a^3*b^2*c + a^2*b^3*c - 2*a*b^4*c -
      2*a^3*b*c^2 - 2*a^2*b^2*c^2 + a*b^3*c^2 - b^4*c^2 + 4*a^3*c^3 +
      6*a^2*b*c^3 + 2*a*b^2*c^3 - b^3*c^3 + a*b*c^4 - 3*b^2*c^4 -
      2*a*c^5 - 3*b*c^5 - a^5*RB - a^4*b*RB + a^3*b^2*RB + a^2*b^3*RB -
      a^4*c*RB - 4*a^3*b*c*RB - a^2*b^2*c*RB + 2*a*b^3*c*RB +
      2*a^3*c^2*RB + 2*a^2*b*c^2*RB - a*b^2*c^2*RB + b^3*c^2*RB +
      2*a^2*c^3*RB + 4*a*b*c^3*RB + b^2*c^3*RB - a*c^4*RB - b*c^4*RB -
      c^5*RB)*(a^4 - 2*a^2*b^2 + b^4 + a^2*c^2 + 2*a*b*c^2 - 3*b^2*c^2 +
      2*c^4 - 2*a^2*c*RC + 2*b^2*c*RC - 2*c^3*RC))}
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