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Locus

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  • Antreas
    Let ABC be a triangle and PaPbPc the pedal triangle of point P. Denote: A B C the triangle bounded by the radical axes:
    Message 1 of 240 , Mar 3, 2013
      Let ABC be a triangle and PaPbPc the pedal triangle of point P.

      Denote:

      A'B'C' the triangle bounded by the radical axes:
      ((circle_with_diameter_OA),(circle_with_diameter_PPa)),
      ((circle_with_diameter_OB),(circle_with_diameter_PPb)),
      ((circle_with_diameter_OC),(circle_with_diameter_PPc)).

      Which is the locus of P such that ABC, A'B'C'
      are perspective?

      H is on the locus. Perspector?

      APH
    • Antreas Hatzipolakis
      [APH]: Let ABC be a triangle. A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.
      Message 240 of 240 , Feb 16

        [APH]:

         

        Let ABC be a triangle.

        A line L passing through A intersects the circle with diameter AB again at Ab and the circle with diameter AC again at Ac.

        [Equivalently: Let Ab, Ac be the orthogonal projections of B, C on L, resp.]
        Let A* be the intersection of BAc and CAb.

        Which is the locus of A* as L moves around A?


        [César Lozada]:


        Parametric trilinear equation:

         

        1/u(t) = a*((b^2+c^2-a^2)^2-4*b^2*c^2*c os(2*t)^2)/(2*S)

         

        1/v(t) = 2*(cos(2*t)*c-b)*S - c*sin(2*t)*(a^2+3*b^2-2*cos(2* t)*b*c-c^2)

         

        1/w(t) = 2*(cos(2*t)*b-c)*S + b*sin(2*t)*(a^2+3*c^2-2*cos(2* t)*b*c-b^2)

         

        Regards,

        César Lozada

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