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

Quadrilateral Problem

Expand Messages
  • Antreas
    An old problem: Let ABCD be a quadrilateral. To construct two congruent and tagent circles such that each one be tagent to sides of opposite angles of ABCD. So
    Message 1 of 4 , Feb 17, 2013
    • 0 Attachment
      An old problem:
      Let ABCD be a quadrilateral. To construct two congruent and tagent circles such that each one be tagent to sides of opposite angles of ABCD.

      So we have two cases:

      (Ka),(Kc) tagent to sides of A,C, resp.

      (Kb),(Kd) tagent to sides of B, D, resp.

      My Question:

      For which quadrilaterals the four centers Ka, Kb, Kc, Kd are
      concyclic? For cyclic ones maybe?

      APH
    • Ignacio Larrosa Cañestro
      ... A synthetic construction for that circles exist, because its radius can be determined from the vertices coordinates by rational operations and square
      Message 2 of 4 , Feb 19, 2013
      • 0 Attachment
        El 18/02/2013 0:26, Antreas escribió:
        > An old problem:
        > Let ABCD be a quadrilateral. To construct two congruent and tagent circles such that each one be tagent to sides of opposite angles of ABCD.
        >
        > So we have two cases:
        >
        > (Ka),(Kc) tagent to sides of A,C, resp.
        >
        > (Kb),(Kd) tagent to sides of B, D, resp.
        >
        > My Question:
        >
        > For which quadrilaterals the four centers Ka, Kb, Kc, Kd are
        > concyclic? For cyclic ones maybe?
        >
        > APH

        A synthetic construction for that circles exist, because its radius can
        be determined from the vertices coordinates by rational operations and
        square roots. But, it's known a simple synthetic one?


        --
        Saludos,

        Ignacio Larrosa Cañestro
        A Coruña (España)
        ilarrosa@...
        http://www.xente.mundo-r.com/ilarrosa/GeoGebra/
      • Antreas
        [APH] ... See the construction of the circles here: http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html Note that we get a quadrilateral
        Message 3 of 4 , Feb 26, 2013
        • 0 Attachment
          [APH]
          > An old problem:
          > Let ABCD be a quadrilateral. To construct two congruent and
          > tagent circles such that each one be tagent to sides of
          > opposite angles of ABCD.
          >
          > So we have two cases:
          >
          > (Ka),(Kc) tagent to sides of A,C, resp.
          >
          > (Kb),(Kd) tagent to sides of B, D, resp.
          >
          > My Question:
          >
          > For which quadrilaterals the four centers Ka, Kb, Kc, Kd are
          > concyclic? For cyclic ones maybe?

          See the construction of the circles here:

          http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html

          Note that we get a quadrilateral center, the point KaKc /\ KbKd,
          and a quadrilateral central line, the line joining the points
          of contact of (Ka),(Kc) and (Kb),(Kd).

          Antreas
        • Ignacio Larrosa Cañestro
          ... From: Antreas To: Sent: Tuesday, February 26, 2013 10:07 AM Subject: [EMHL] Re: Quadrilateral Problem
          Message 4 of 4 , Feb 26, 2013
          • 0 Attachment
            ----- Original Message -----
            From: "Antreas" <anopolis72@...>
            To: <Hyacinthos@yahoogroups.com>
            Sent: Tuesday, February 26, 2013 10:07 AM
            Subject: [EMHL] Re: Quadrilateral Problem


            > [APH]
            >> An old problem:
            >> Let ABCD be a quadrilateral. To construct two congruent and
            >> tagent circles such that each one be tagent to sides of
            >> opposite angles of ABCD.
            >>
            >> So we have two cases:
            >>
            >> (Ka),(Kc) tagent to sides of A,C, resp.
            >>
            >> (Kb),(Kd) tagent to sides of B, D, resp.
            >>
            >> My Question:
            >>
            >> For which quadrilaterals the four centers Ka, Kb, Kc, Kd are
            >> concyclic? For cyclic ones maybe?
            >
            > See the construction of the circles here:
            >
            > http://anthrakitis.blogspot.gr/2013/02/circles-of-quadrilateral.html



            Thanks. I make an applet with GeoGebra:

            http://www.xente.mundo-r.com/ilarrosa/GeoGebra/DosCircTg4l.html

            based on a construction of Eduardo in es.ciencia.matematicas.


            > Note that we get a quadrilateral center, the point KaKc /\ KbKd,
            > and a quadrilateral central line, the line joining the points
            > of contact of (Ka),(Kc) and (Kb),(Kd).
            >
            > Antreas
            >
            >
            >
          Your message has been successfully submitted and would be delivered to recipients shortly.