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Re: [EMHL] NPC locus

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  • Boutte Gilles
    Dear Antreas, Paul and other Hyacinthists ... Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA have the same NPC center N. A is the
    Message 1 of 5 , Oct 3, 2001
      Dear Antreas, Paul and other Hyacinthists

      > [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so
      > that the NPC center of ABC lies on AB ?

      Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA have
      the same NPC center N.

      A is the orthocenter of HBC, thus "N lies on AB" means "N lies on an
      altitude of HBC".

      N lies on an altitude iff this altitude is the euler line of the triangle
      iff the triangle is isosceles.

      N lies on AB iff BH=BC.

      The locus of H is the circle with center B and radius BC.
      The locus of A is the locus of the orthocenter of HBC :

      an orthostrophoid with double point at B and asymptot perpendicular to BC at
      C' reflection of C about B.

      Friendly

      Gilles
    • Paul Yiu
      Dear Gilles and Antreas, [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so that the NPC center of ABC lies on AB ? ^^ [PY]: The locus is
      Message 2 of 5 , Oct 3, 2001
        Dear Gilles and Antreas,

        [APH]: Let ABC be a triangle. If BC is fixed, which is the locus of A so that
        the NPC center of ABC lies on AB ?
        ^^
        [PY]: The locus is the rectangular hyperbola with vertices B and C.

        [GB]: Let H be the orthocenter of ABC: the four triangles ABC, HAB, HBC,HCA
        have the same NPC center N.

        A is the orthocenter of HBC, thus "N lies on AB" means "N lies on an
        altitude of HBC".

        N lies on an altitude iff this altitude is the euler line of the triangle iff
        the triangle is isosceles.

        N lies on AB iff BH=BC.

        The locus of H is the circle with center B and radius BC.
        The locus of A is the locus of the orthocenter of HBC :

        an orthostrophoid with double point at B and asymptot perpendicular to BC at
        C' reflection of C about B.

        ****
        Oh! I see. I had solved a different but easier problem: I thought Antreas was
        asking for the locus of A for which the NPC of ABC lies on BC. Now I see he
        wrote AB.

        Best regards
        Sincerely
        Paul
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