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Locus

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  • Antreas Hatzipolakis
    Let ABC be a triangle, P a point and (a),(Ob),(Oc) the circles with diameters PA,PB,PC, resp. The circles (Ob), (Oc) intersect in point A1, other than P, on BC
    Message 1 of 234 , Feb 28, 2013
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      Let ABC be a triangle, P a point and (a),(Ob),(Oc) the circles
      with diameters PA,PB,PC, resp.

      The circles (Ob), (Oc) intersect in point A1, other than P, on BC
      Similarly B1,C1. The triangle A1B1C1 is the pedal triangle of P.

      Let A2B2C2 be the cevian triangle of P.

      The circles (Oa),(Ob), (Oc) intersect (O) at A3,B3,C3 [other than A,B,C, resp.]

      The lines AA3, BB3, CC3 [= readical axes of
      ((O),((Oa),((O),((Ob),((O),((Oc), resp.]
      bound a triangle A4B4C4.

      Loci:

      Which is the locus of P such that be perspective the triangles:

      1. ABC, A1B1C1 : it is Darboux cubic (since A1B1C1 is the pedal triangle of P)

      2. ABC, A4B4C4

      3. A1B1C1, A3B3C3

      4. A1B1C1, A4B4C4

      5. A2B2C2, A3B3C3,

      6. A2B2C2, A4B4C4

      7. A3B3C3, A4B4C4

      I think that in all cases the locus is Darboux cubic + possibly something else.

      APH
    • Antreas Hatzipolakis
      ... [Telv Cohl] Pedal curve http://en.wikipedia.org/wiki/Pedal_curve -- http://anopolis72000.blogspot.com/
      Message 234 of 234 , Dec 4, 2014
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        [APH]:
        Let (O) be a fixed circle, Q a fixed point and P a point on the circle.

        Let Qp be the orthogonal projection of Q on the tangent to circle at P.

        Which is the locus of Qp as P moves on the circle?

        Generalization: (O) = a fixed conic.

        aph



        [Telv Cohl]

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