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Re: GENERALIZATION (was: 2003 LAST CONJECTURE !)

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  • jpehrmfr
    Dear Antreas ... In both cases, your assertions are true and the locus of the common point of the three NP-circles is the line joining H to X(143) = the
    Message 1 of 8 , Jan 1, 2004
      Dear Antreas
      > Let ABC be a triangle, and A'B'C' its Orthic Triangle.
      >
      > 1. Let Pa, Pb, Pc be points on AH, BH, CH, resp. such that:
      >
      > APa / AH = BPb / BH = CPc / CH = t
      >
      > Denote
      >
      > Ab, Ac = the reflections of Pa in CC', BB', resp.
      > Bc, Ba = the reflections of Pb in AA', CC', resp.
      > Ca, Cb = the reflections of Pc in BB', AA', resp.
      >
      > The Nine Point Circles of the Triangles
      > PaAbAc, PbBcBa, PcCaCb are concurrent (??)
      >
      > Which is the locus of the point P of concurrence,
      > as t varies?
      >
      > [If t = 0 (ie Pa = A, Pb = B, Pc = C), then P = X(1986) (BW)]
      >
      > 2. Let Pa, Pb, Pc be points on A'H, B'H, C'H, resp. such that:
      >
      > A'Pa / A'H = B'Pb / B'H = C'Pc / C'H = t'
      >
      > Denote
      >
      > Ab, Ac = the reflections of Pa in CC', BB', resp.
      > Bc, Ba = the reflections of Pb in AA', CC', resp.
      > Ca, Cb = the reflections of Pc in BB', AA', resp.
      >
      > The Nine Point Circles of the Triangles
      > PaAbAc, PbBcBa, PcCaCb are concurrent (??)
      >
      > Which is the locus of the point P of concurrence,
      > as t' varies?
      >
      > [If t' = 0 (ie Pa = A', Pb = B', Pc = C'), then P = X(1112) (JPE) ]

      In both cases, your assertions are true and the locus of the common
      point of the three NP-circles is the line joining H to X(143) = the
      NP-center of the orthic triangle

      Friendly. Jean-Pierre
    • jpehrmfr
      Dear Antreas ... They go through X(974) Happy and peaceful 2004 to every Hyacinthist Friendly. Jean-Pierre
      Message 2 of 8 , Jan 1, 2004
        Dear Antreas
        > Let A'B'C' be the orthic and A"B"C" the medial
        > triangles of ABC.
        >
        > Denote:
        >
        > Ab, Ac = the reflections of A" in CC', BB', resp.
        > Bc, Ba = the reflections of B" in AA', CC', resp.
        > Ca, Cb = the reflections of C" in BB', AA', resp.
        >
        > CONJECTURE:
        >
        > The Nine Point Circles of the triangles
        > A"AbAc, B"BcBa, C"CaCb concur.

        They go through X(974)
        Happy and peaceful 2004 to every Hyacinthist
        Friendly. Jean-Pierre
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