Loading ...
Sorry, an error occurred while loading the content.

Re: [EMHL] NPC locus

Expand Messages
  • Boutte Gilles
    Dear Paul ... Construct an equilateral triangle HBC having base BC. Let A be its orthocenter. The four triangles ABC, HBC, HAB, HAC have the same NPC center
    Message 1 of 5 , Oct 3, 2001
    • 0 Attachment
      Dear Paul

      > [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]: This is the rectangular hyperbola with B and C as vertices.

      Construct an equilateral triangle HBC having base BC. Let A be its
      orthocenter. The four triangles ABC, HBC, HAB, HAC have the same NPC center
      : A, the orthocenter of the equilateral triangle HBC, which lies on the
      perpendicular bisector of BC, but not on the rectangular hyperbola with B
      and C as vertices.

      Friendly

      Gilles
    • 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 2 of 5 , Oct 3, 2001
      • 0 Attachment
        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 3 of 5 , Oct 3, 2001
        • 0 Attachment
          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
        Your message has been successfully submitted and would be delivered to recipients shortly.