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21052Re: Simson lines

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  • Daniel
    Jun 6, 2012
      Yes, it is a particular case of S-triangles.

      Two triangles ABC and A'B'C' inscribed in the same circle are S-triangles iff arc(AA')+arc(BB')+arc(CC')=0 (mod 2PI).
      The Simson lines of A',B',C', with respect to ABC and the Simson lines of A,B,C, with respect to A'B'C' are concurrent in the midpoint of [HH'] where H and H' are the orthocenters of ABC and A'B'C'.

      The triangles CAB and CPQ are S-triangles iff PQ is parallel to AB.
      So, the Simson lines of P,Q, and C are concurrent. But the Simson line of C is the altitude from C.

      See
      [1] Traian Lalescu, A Class of Remarkable Triangles,
      Gazeta Matematica , vol XX, feb. 1915 ,p 213
      (in Romanian)

      [2] Trajan Lalesco, La geometrie du triangle ,Bucharest,
      1937, Paris, Libraire Vuibert

      Sincerely
      Daniel Vacaretu


      --- In Hyacinthos@yahoogroups.com, "Alexey Zaslavsky" <zasl@...> wrote:
      >
      > Dear colleagues!
      > Is next fact known?
      > Let P, Q be two point on the circumcircle of triangle ABC. Then their Simson lines meet on the altitude from C iff PQ is parallel to AB.
      >
      > Sincerely Alexey
      >
      > [Non-text portions of this message have been removed]
      >
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