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Locus

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  • Antreas Hatzipolakis
    Let ABC be a triangle, IaIbIc the excentral triangle, P a point and A B C the cevian triangle of P. Denote A* = B Ic / C Ib, B* = C Ia / A Ic, C* = A Ib /
    Message 1 of 240 , Feb 1, 2013
      Let ABC be a triangle, IaIbIc the excentral triangle, P a point
      and A'B'C' the cevian triangle of P.

      Denote A* = B'Ic /\ C'Ib, B* = C'Ia /\ A'Ic, C* = A'Ib /\ B'Ia

      Which is the locus of P such that

      1. ABC, A*B*C*

      2. A'B'C', A*B*C*

      are perspective?

      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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