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## Effect of Projection on Homogeneous Barycentric Coordinates

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• 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.)

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