## orthojoin

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• In ETC points 1512 through 1568 are the orthojoin of various points This whole section seems irrelevant to anything (most are what I call lonely points
Message 1 of 4 , Mar 3, 2006
In ETC points 1512 through 1568 are the "orthojoin" of various points

This whole section seems irrelevant to anything (most are what I call
"lonely points" which little connection to other points or lines).

For some points (such as the Gossard point) their lonliness is a
descriptive quality. For these points it seems to indicate their
irrelevance.

I call them "silly points."

I plan to drop them from my next pictures of the distribution of
triangle points.

Would anyone out there like to defend these points or the orthojoin
operation.

Steve
• ... points ... call ... orthojoin ... Dear Steve, The orthojoin stems from the orthopole which seems a neat geometric construct. I am hesitant to dismiss the
Message 2 of 4 , Mar 3, 2006
> In ETC points 1512 through 1568 are the "orthojoin" of various
points
>
>
> This whole section seems irrelevant to anything (most are what I
call
> "lonely points" which little connection to other points or lines).
>
> For some points (such as the Gossard point) their lonliness is a
> descriptive quality. For these points it seems to indicate their
> irrelevance.
>
> I call them "silly points."
>
> I plan to drop them from my next pictures of the distribution of
> triangle points.
>
> Would anyone out there like to defend these points or the
orthojoin
> operation.
>
> Steve
>

Dear Steve,

The orthojoin stems from the orthopole which seems a neat geometric
construct. I am hesitant to dismiss the orthojoin. The coordinates
given in ETC are quite ugly, but that should not detract from the
possible uses of the operation.

Sincerely, Jeff
• ... Jef, So what are the possible uses? A clear thought process is based on good fundamentals. I suspect that the orthojoin is not one of these. Look at how
Message 3 of 4 , Mar 4, 2006
On Mar 4, 2006, at 1:01 AM, Jeff Brooks wrote:

> The orthojoin stems from the orthopole which seems a neat geometric
> construct. I am hesitant to dismiss the orthojoin. The coordinates
> given in ETC are quite ugly, but that should not detract from the
> possible uses of the operation.
>
> Sincerely, Jeff

Jef,

So what are the possible uses?

A clear thought process is based on good fundamentals. I suspect that
the orthojoin is not one of these.

Look at how algebra has developed. There are only a few operations
out of which more complicated operations can be expressed. When we
wish a complicated operation, we invent a function on an ad hoc
basis. Then we forget that function as in other contexts.

But I am suspicious that the orthojoin dismisses itself. Except for
ad hoc uses, I suspect I will not miss much by not learning this
operation. The Steiner inverse is fundamental. One misses a lot by
not adding it to one's repetoire, but, again except for ad hoc uses,
I suspect that orthojoin is not.

I would love to be convinced otherwise.

Regards to OK from the Eastern US,

Steve

[Non-text portions of this message have been removed]
• ... geometric ... coordinates ... that ... for ... uses, ... Hey Steve, Please let me digress a bit before turning back to the orthojoin operation. I ve been
Message 4 of 4 , Mar 5, 2006
--- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:
>
>
> On Mar 4, 2006, at 1:01 AM, Jeff Brooks wrote:
>
> > The orthojoin stems from the orthopole which seems a neat
geometric
> > construct. I am hesitant to dismiss the orthojoin. The
coordinates
> > given in ETC are quite ugly, but that should not detract from the
> > possible uses of the operation.
> >
> > Sincerely, Jeff
>
> Jef,
>
> So what are the possible uses?
>
> A clear thought process is based on good fundamentals. I suspect
that
> the orthojoin is not one of these.
>
> Look at how algebra has developed. There are only a few operations
> out of which more complicated operations can be expressed. When we
> wish a complicated operation, we invent a function on an ad hoc
> basis. Then we forget that function as in other contexts.
>
> But I am suspicious that the orthojoin dismisses itself. Except
for
> ad hoc uses, I suspect I will not miss much by not learning this
> operation. The Steiner inverse is fundamental. One misses a lot by
> not adding it to one's repetoire, but, again except for ad hoc
uses,
> I suspect that orthojoin is not.
>
> I would love to be convinced otherwise.
>
> Regards to OK from the Eastern US,
>
> Steve
>
Hey Steve,

Please let me digress a bit before turning back to the orthojoin
operation.

I've been thinking a lot lately about how different geometric
constructions yield the same function and conversely how different
functions can provide the same geometric meaning. As you know, there
was some discussion recently about Keith Dean and Floor van Lamoen's
paper on "Geometric Construction of Reciprocal Conjugations." I read
with great interest the different approaches each person took in
trying to help me to better understand the development of this
generalized conjugation. One person pointed out that a particular
step was used in the authors' paper to simply `motivate' the
development of the subsequent material. I was also provided some
particular step. I truly understand these sentiments especially in
light of my poorly worded questions and comments.

To quote Henri Poincarẻ "Mathematicians do not study objects, but
relations among objects; they are indifferent to the replacement of
objects by others as long as relations do not change. Matter is not
important, only form interests them."

And, perhaps, this is where I messed up. I did not explain that I
was trying to explore a particular form. I honestly don't care
whether a proof is given analytically or synthetically because I know
that I can work from either. Having said this, let me also state
that it is typically easier for me to follow along with pictures of
some sort.

For example, I have constructed the inverse in the circumcircle using
basic geometric methods and I now have a `feel' for this operation
that I can relate to similar problems or situations. I know of
several different formulas for the inverse in the circumcircle, but I
don't know why Clark Kimberling chose to use Peter Moses'
formulations to highlight some common inverses given in the
Encyclopedia of Triangle Centers. I suspect he has a reason.

Likewise, I am confident that I can use Antreas' theorem given in
Hyacinthos message #11327 to construct points of the form P/(X U)
because I have a lot of evidence to support this conjecture. This
seems more general than the definition of reciprocal conjugate. But
even if I prove this, I would still be left wondering about that
basic step I missed earlier in Floor and Keith's paper. I would also
be left wondering how this might relate to the discussions you and