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

Reflections and Parallels (Locus)

Expand Messages
  • Antreas P. Hatzipolakis
    Let ABC be a triangle, P, P* two isogonal points, A B C the pedal triangle of P, and A*B*C* the cevian triangle of P*. Let A be the reflection of A* in A ,
    Message 1 of 5 , Aug 28, 2005
    • 0 Attachment
      Let ABC be a triangle, P, P* two isogonal points,
      A'B'C' the pedal triangle of P, and A*B*C* the
      cevian triangle of P*.

      Let A" be the reflection of A* in A', and La the parallel
      to AA* through A". Similarly La,Lc.

      Which is the locus of P such that La,Lb,Lc are concurrent?
      Locus of the point of concurrence?

      Greetings

      Antreas

      --
    • Antreas P. Hatzipolakis
      ... I think that the lines La,Lb,Lc are ALWAYS concurrent (ie the locus is the whole plane). Is it true??? Antreas --
      Message 2 of 5 , Aug 29, 2005
      • 0 Attachment
        [APH]:

        >Let ABC be a triangle, P, P* two isogonal points,
        >A'B'C' the pedal triangle of P, and A*B*C* the
        >cevian triangle of P*.
        >
        >Let A" be the reflection of A* in A', and La the parallel
        >to AA* through A". Similarly La,Lc.
        >
        >Which is the locus of P such that La,Lb,Lc are concurrent?
        >Locus of the point of concurrence?


        I think that the lines La,Lb,Lc are ALWAYS concurrent
        (ie the locus is the whole plane).

        Is it true???


        Antreas
        --
      • Nikolaos Dergiades
        Yes Antreas it is true and the lines are parallel if P lies on the circumcircle. It is more easy to prove first that the lines from A , B , C parallel to the
        Message 3 of 5 , Aug 30, 2005
        • 0 Attachment
          Yes Antreas it is true
          and the lines are parallel if P lies on the circumcircle.

          It is more easy to prove first that the lines
          from A', B', C' parallel to the lines AA*,BB*,CC*
          are ALWAYS concurrent and if Q is the point of
          concurrence then your point of concurrence is the
          reflection of P* in Q.

          Best regards
          Nikolaos Dergiades

          > [APH]:
          >
          > >Let ABC be a triangle, P, P* two isogonal points,
          > >A'B'C' the pedal triangle of P, and A*B*C* the
          > >cevian triangle of P*.
          > >
          > >Let A" be the reflection of A* in A', and La the parallel
          > >to AA* through A". Similarly La,Lc.
          > >
          > >Which is the locus of P such that La,Lb,Lc are concurrent?
          > >Locus of the point of concurrence?
          >
          >
          > I think that the lines La,Lb,Lc are ALWAYS concurrent
          > (ie the locus is the whole plane).
          >
          > Is it true???
          >
          >
          > Antreas
          >
          >
        • Antreas P. Hatzipolakis
          ... Dear Nikos [welcome back from summer vacation!] I have parametrized it. See ANOPOLIS : http://groups.yahoo.com/group/Anopolis/ message #94 BTW, I added a
          Message 4 of 5 , Aug 30, 2005
          • 0 Attachment
            >Yes Antreas it is true
            >and the lines are parallel if P lies on the circumcircle.
            >
            >It is more easy to prove first that the lines
            >from A', B', C' parallel to the lines AA*,BB*,CC*
            >are ALWAYS concurrent and if Q is the point of
            >concurrence then your point of concurrence is the
            >reflection of P* in Q.


            Dear Nikos
            [welcome back from summer vacation!]

            I have parametrized it.

            See ANOPOLIS : http://groups.yahoo.com/group/Anopolis/
            message #94

            BTW, I added a photograph in the group.
            (Einai h tampela tou xwriou mou, diatrhth apo tis
            mpalw8ies twn xwrianwn mou !!)

            I think it is a good idea to add a photograph
            in hyacinthos group as well. Maybe the photograph
            of Emile Lemoine (or one of mine! :-))

            Antreas


            >Best regards
            >Nikolaos Dergiades
            >
            >> [APH]:
            >>
            >> >Let ABC be a triangle, P, P* two isogonal points,
            >> >A'B'C' the pedal triangle of P, and A*B*C* the
            >> >cevian triangle of P*.
            >> >
            >> >Let A" be the reflection of A* in A', and La the parallel
            >> >to AA* through A". Similarly La,Lc.
            >> >
            >> >Which is the locus of P such that La,Lb,Lc are concurrent?
            >> >Locus of the point of concurrence?
            >>
            >>
            >> I think that the lines La,Lb,Lc are ALWAYS concurrent
            >> (ie the locus is the whole plane).
            >>
            >> Is it true???
            >>
            >>
            >> Antreas
            >>
            >>
            >
            >
            >
            >
            >Yahoo! Groups Links
            >
            >
            >
            >

            --
          • peter_mows
            Dear Antreas & Nikolaos, ... [ND] ... [APH] ... Think that .. Locus of the points of concurrence is the line connecting P* and Q. The point of concurrence, pC,
            Message 5 of 5 , Aug 30, 2005
            • 0 Attachment
              Dear Antreas & Nikolaos,

              [APH]:
              >Let ABC be a triangle, P, P* two isogonal points,
              >A'B'C' the pedal triangle of P, and A*B*C* the
              >cevian triangle of P*.
              >
              >Let A" be the reflection of A* in A', and La the parallel
              >to AA* through A". Similarly La,Lc.
              >
              >Which is the locus of P such that La,Lb,Lc are concurrent?
              >Locus of the point of concurrence?
              >
              > I think that the lines La,Lb,Lc are ALWAYS concurrent
              > (ie the locus is the whole plane).
              >
              > Is it true???

              [ND]
              >Yes Antreas it is true
              >and the lines are parallel if P lies on the circumcircle.
              >
              >It is more easy to prove first that the lines
              >from A', B', C' parallel to the lines AA*,BB*,CC*
              >are ALWAYS concurrent and if Q is the point of
              >concurrence then your point of concurrence is the
              >reflection of P* in Q.


              [APH]
              >I have parametrized it.
              >
              >Let ABC be a triangle, P, P* two isogonal points,
              >A'B'C' the pedal triangle of P, and A*B*C* the
              >cevian triangle of P*.

              >Let A", B", C" be points on BC, CA, AB resp. such that

              >A"A' / A"A* = B"B' / B"B* = C"C' / C"C* = t,

              >and La,Lb,Lc parallels to AA*,BB*,CC* through A",B",C", resp.

              >Are the lines La,Lb,Lc concurrent for all t's ?

              >Locus of point of concurrence as t varies?

              Think that ..
              Locus of the points of concurrence is the line connecting P* and Q.

              The point of concurrence, pC, is such that |pCQ| / |pCP*| = t

              Best regards,
              Peter.
            Your message has been successfully submitted and would be delivered to recipients shortly.