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RE: [EMHL] Locus

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  • Paul Yiu
    Dear Antreas, Let ABC be a triangle, P = (u:v:w) with cevian triangle A B C and Q=(x:y:z) with anticevian triangle QaQbQc. Denote A* = B Qc / C Qb, B* =
    Message 1 of 240 , Feb 1, 2013
      Dear Antreas,

      Let ABC be a triangle, P = (u:v:w) with cevian triangle A'B'C' and Q=(x:y:z) with anticevian triangle QaQbQc.

      Denote A* = B'Qc /\ C'Qb, B* = C'Qa /\ A'Qc, C* = A'Qb /\ B'Qa.

      A*B*C* is perspective with

      (1) ABC at ( x/(-x/u+y/v+z/w) : y/(x/u-y/v+z/w) : z/(x/u+y/v-z/w)),
      (2) A'B'C' at Q,
      (3) QaQbQc at (x^2/u : y^2/v: z^2/w).

      Note: A'B'C' and QaQbQc are perspective at the cevian quotient
      P/Q = ( x(-x/u+y/v+z/w) : y(x/u-y/v+z/w) : z(x/u+y/v-z/w)).

      Best regards
      Sincerely
      Paul
    • 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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