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