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Effect of Projection on Homogeneous Barycentric Coordinates

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  • bbblow
    Hello everyone. I ve been studying the effect of projection on homogeneous barycentric coordinates, to generalize some theorems in triangle geometry
    Message 1 of 1 , Sep 6, 2010
      Hello everyone.

      I've been studying the effect of projection on homogeneous barycentric
      coordinates, to generalize some theorems in triangle geometry
      projectively.

      The projection, turns out to have a very simple and interesting effect
      on the coordinates. Using preservation of Cross-Ratio, one can easily
      see that



      When the centroid G(1:1:1) is mapped to ¥Õ(p:q:r), arbitrary point
      (x:y:z) is mapped to its barycentric product of
      ¥Õ¡¿(x:y:z)=(px:qy:rz).



      I'd like to present a generalized version of Feuerbach's Theorem (9point
      circle is tangent to incircles and excircles) using the above formula
      regarding projection.

      Since we can see nine-point circle as the bicevian conic of G and H, we
      see that the projective generalization of Feuerbach's Theorem is:



      For given arbitrary points ¥Õ, I, let Ia, Ib, Ic be the cevian
      associates of I.

      Take the isotomic of anticomplement of each of I, Ia, ... and name them
      J, Ja, ...

      Take the isotomic of anticomplement of I¡¿I (barycentric square), and
      name it H.

      Take the bicevian conic for ¥Õ, ¥Õ¡¿H, name it ¥Ã9, and take
      inconics with perspectors ¥Õ¡¿J, ¥Õ¡¿Ja, ... and name them ¥Ã,
      ¥Ãa, ...

      Then ¥Ã9 is tangent to ¥Ã, ¥Ãa, ¥Ãb, ¥Ãc.



      Note that I is an arbitrary point here because we are talking of
      incenter of pre-projective triangle.

      Using the above generalization, we can obtain a bicevian conic tangent
      to a given inconic too:



      For inconic with perspector (u:v:w), a bicevian conic correspondent to
      the following two points is tangent to the inconic:

      (x:y:z)

      (x/((x/u)(x/u + y/v + z/w) - yz/vw ) : ...)



      where (x:y:z) can be any point.



      I'd like to know if anyone knows whether this is a new result. Was
      discovered before (or maybe something so trivial that it is not worth to
      be called 'discovery', haha.)



      [Non-text portions of this message have been removed]
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