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## ex-extra perspectors

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• Dear Hyacinthers, A question: How can I construct the triangle (extra-operation) PaPbPc from P =Po ? A remark: Rouché-Comberousse (Note III sur la géométrie
Message 1 of 2 , Sep 6, 2000
Dear Hyacinthers,
A question: How can I construct the triangle (extra-operation) PaPbPc
from P =Po ?
A remark: Rouché-Comberousse (Note III sur la géométrie récente du
triangle, traité de géométrie 1912)
speak about "points associés" instead of "precevians points" and "points
adjoints" instead of "extra-points".

Regards
Fred Lang
Switzerland
• ... You can t construct Pa Pb Pc from Po, because they aren t functions of Po. Rather, they are the algebraic conjugates of the functions that were used
Message 2 of 2 , Sep 9, 2000
On Wed, 6 Sep 2000, Fred Lang wrote:

> A question: How can I construct the triangle (extra-operation) PaPbPc
> from P =Po ?
> A remark: Rouché-Comberousse (Note III sur la géométrie récente du
> triangle, traité de géométrie 1912)
> speak about "points associés" instead of "precevians points"
> and "points adjoints" instead of "extra-points".

You can't construct Pa Pb Pc from Po, because they
aren't functions of Po. Rather, they are the algebraic
conjugates of the functions that were used to define Po.

Let's consider, for instance,

the incenter Io = (a:b:c),
and
the Spieker point So = (b+c:c+a:a+b).

For these, b-extraversion yields

Ib = (a:-b:c) and Sb = (c-b:c+a:a-b).

Now for an equilateral triangle both Io and So are
the center, but we have

Ib = (1:-1:1) , Sb = (0:2:0),

which are different.

The extraversion operations are not defined on points,
but rather, on constructions for points.

I'm glad to hear your remark, which confirms that the
standard term for A^P, B^P, C^P is "harmonic associates"
(of P). I have not previously seen a term for extraversion
in any published work, and so am glad to hear of "points
adjoints".

My guess is that it doesn't mean precisely the same thing,
because there are several closely related notions here, from
which I chose the particular one I call "extraversion" only
after considerable thought.

John Conway
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