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Vecten again

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  • xpolakis
    Locus: Let (Q) be a circle, A a point on the circle, and BC a fixed line segment. Which is the locus of a point P of ABC, as A moves on the circle? If P = O,
    Message 1 of 1 , Apr 1, 2009
      Locus:

      Let (Q) be a circle, A a point on the circle, and BC a fixed line
      segment.

      Which is the locus of a point P of ABC, as A moves on the circle?

      If P = O, the locus is the perpendicular bisector of BC.

      If P = G,H,N,I the locus is some conic.
      Let's denote its center as A(P)
      [ A(G), A(H), A(N), A(I) for P = G,H,N,I]

      Application:

      Let's apply the above to Vecten's configuration:

      Let ABC be a triangle, and BCBaCa, CACbAb, ABAcBc.
      the Vecten's squares.

      Now, take the circumcircle (O) of ABC (as (Q) circle in the locus
      problem)and the segments BaCa, CbAb, AcBc (as the segment BC in
      the locus problem),the vertices A,B,C and the centers of the loci
      A(P), B(P), C(P).

      Is the triangle ABC perspective with A(P)B(P)C(P)
      for some P = G,H,N,I ?

      Of course we can apply the locus problem to any other
      well-defined triad of fixed line segments of ABC (its
      sides included).

      APH
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