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16516Re: Simson lines

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  • jpehrmfr
    Jul 1 2:18 AM
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      Dear Alexei
      > The polar transformation and the properties of Simson lines allow to
      prove next interesting fact.
      > Let I be the incenter of ABC. For an arbitrary point P we denote as
      P_A the point on the line PA such that IA is the bisector (internal or
      external) of angle PIP_A. The points P_B, P_C are defined similarly.
      > 1. If P_A, P_B, P_C are collinear then this line touches the incircle
      of ABC.
      > 2. The locus of points P with this property is the line.
      > Can anybody prove this without using the polar transformation?

      Here is a better presentation of a generalization :
      Starting with a point J in the plane of ABC, the perpendicular lines
      through J to JA, JB, JC intersect BC, CA, AB respectively at A', B', C'.
      It is well known that A',B',C' are on a same line L(J).
      For any point M,
      Ma is the point of the line AM such as <MJA = <AJMa
      Mb is the point of the line BM such as <MJB = <BJMb
      Mc is the point of the line CM such as <MJC = <CJMc
      Then Ma,Mb,Mc are collinear if and only if M lies on L(J)
      In this case, the four lines A'Ma, B'Mb, C'Mc, MaMbMc touch the
      inscribed conic with focus J
      This can be proved with projective tools using the fact that M->Ma is
      the harmonic homology with center A and axis the line JA'
      * The harmonic homology (I don't know if this English name is correct)
      with center O, axis L maps M to his harmonic conjugate wrt O and OM
      inter L

      I think that, if we consider three harmonic homologies with centers
      A,B,C, the locus of M with three colinear images is generally a cubic,
      even when the three axis concur. In the particular case of three axis
      La,Lb,Lc going through a point J and such as there exists an involution
      swapping (JA,La), (JB,Lb),(JC,Lc), then the cubic degenerates in three
      lines :
      the fixed lines of the involution and the line through La inter BC, Lb
      inter CA, Lc inter AB
      Friendly. Jean-Pierre
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