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10192Re: More Orth. Proj. of H (was: IMO 2004 Problem 1 revisited)

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  • Paul Yiu
    Aug 2, 2004
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      Dear Antreas,

      [APH]: Let ABC be a triangle, P a point, A', B', C' the orth.
      projections of H on AP, BP, CP, resp. and Ea, Eb, Ec
      the midpoints of AH, BH, CH (ie EaEbEc = the Euler
      triangle of ABC)

      Which is the locus of P such that A'B'C', EaEbEc
      are perspective? (Locus of the perspectors?)

      I think that I, N are points of the locus.

      *** Yes, the locus the quintic

      cyclic sum a^2(S_{CA}+S_{AB}+2S_{BC})v^2w^2(S_Bv - S_Cw)
      + cyclic uvw S_A(S_B-S_C)(b^2c^2u^2- a^2S_Avw) = 0.

      It also contains H (trivially), X_{80}, and X_{1263}. Note that X(80)
      and X(1263) are respectively the reflection conjugates of I and N

      I have verified that the quintic is indeed invariant reflection
      conjugation. [The reflection conjugate of a point P is the common
      point of the reflections of the circles PBC, PCA, PAB in the sidelines
      BC, CA, AB respectively].

      Best regards
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