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• Dear Alexey, ... and CD, AC and BD, AD and BC, P - arbitrary point distinct from X, Y, Z. Then the polars of P wrt all conics passing through A, B, C, D have
May 12, 2005 1 of 6
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Dear Alexey,

In message 11252, you wrote:

> Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
and CD, AC and BD, AD and BC, P - arbitrary point distinct from X, Y,
Z. Then the polars of P wrt all conics passing through A, B, C, D
have the common point P'. If A, B, C, D are orthocentric then P' is
isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
is the centroid of three other, then P' is isotomic conjugated. There
is an interesting corollary. Let U, U' and V, V' are two pairs of
conjugated points. Then the points UV^U'V' and U'V^UV' are conjugated.
>

I believe a direct proof of your statements can be given without
referring to well known geometric theorems. Personally, I would love
to see a proof given using complex coordinates...Any ideas?

Sincerely,

Jeff
• Alexey, Of course, I meant message 11251. Sorry, Jeff ... Y, ... There ... conjugated. ... love
May 12, 2005 1 of 6
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Alexey,

Of course, I meant message 11251.

Sorry,

Jeff

>
> In message 11252, you wrote:
>
> > Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
> and CD, AC and BD, AD and BC, P - arbitrary point distinct from X,
Y,
> Z. Then the polars of P wrt all conics passing through A, B, C, D
> have the common point P'. If A, B, C, D are orthocentric then P' is
> isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
> is the centroid of three other, then P' is isotomic conjugated.
There
> is an interesting corollary. Let U, U' and V, V' are two pairs of
> conjugated points. Then the points UV^U'V' and U'V^UV' are
conjugated.
> >
>
> I believe a direct proof of your statements can be given without
> referring to well known geometric theorems. Personally, I would
love
> to see a proof given using complex coordinates...Any ideas?
>
> Sincerely,
>
> Jeff
• Alexey, Doesn t look like we ll have anyone taking up this proposition anytime soon. How about we limit our observations to orthocentric configurations or
May 20, 2005 1 of 6
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Alexey,

Doesn't look like we'll have anyone taking up this proposition
anytime soon. How about we limit our observations to orthocentric
configurations or maybe generalized circumconics just to get the ball
rolling.

Jeff

> Dear Alexey,
>
> In message 11252, you wrote:
>
> > Let give 4 points A, B, C, D. X, Y, Z are the common points of AB
> and CD, AC and BD, AD and BC, P - arbitrary point distinct from X,
Y,
> Z. Then the polars of P wrt all conics passing through A, B, C, D
> have the common point P'. If A, B, C, D are orthocentric then P' is
> isogonally conjugated to P wrt XYZ, and if one of points A, B, C, D
> is the centroid of three other, then P' is isotomic conjugated.
There
> is an interesting corollary. Let U, U' and V, V' are two pairs of
> conjugated points. Then the points UV^U'V' and U'V^UV' are
conjugated.
> >
>
> I believe a direct proof of your statements can be given without
> referring to well known geometric theorems. Personally, I would
love
> to see a proof given using complex coordinates...Any ideas?
>
> Sincerely,
>
> Jeff
• Dear Keith Dean and Floor van Lamoen, Hope one of you are listening. There is a ball headed in your general direction with the word reciprocal conjugation or
May 22, 2005 1 of 6
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Dear Keith Dean and Floor van Lamoen,

Hope one of you are listening. There is a ball headed in your general
direction with the word "reciprocal conjugation" or "isoconjugation"
written on it. I would really appreciate any input you might have
regarding possible solutions to this problem (especially solutions
using complex coordinates).

Sincerely,
Jeff Brooks

PS
Re: http://forumgeom.fau.edu/FG2001volume1/FG200116.pdf

> Alexey,
>
> Doesn't look like we'll have anyone taking up this proposition
> anytime soon. How about we limit our observations to orthocentric
> configurations or maybe generalized circumconics just to get the
ball
> rolling.
>
> Jeff
>
>
>
> > Dear Alexey,
> >
> > In message 11252, you wrote:
> >
> > > Let give 4 points A, B, C, D. X, Y, Z are the common points of
AB
> > and CD, AC and BD, AD and BC, P - arbitrary point distinct from
X,
> Y,
> > Z. Then the polars of P wrt all conics passing through A, B, C, D
> > have the common point P'. If A, B, C, D are orthocentric then P'
is
> > isogonally conjugated to P wrt XYZ, and if one of points A, B, C,
D
> > is the centroid of three other, then P' is isotomic conjugated.
> There
> > is an interesting corollary. Let U, U' and V, V' are two pairs of
> > conjugated points. Then the points UV^U'V' and U'V^UV' are
> conjugated.
> > >
> >
> > I believe a direct proof of your statements can be given without
> > referring to well known geometric theorems. Personally, I would
> love
> > to see a proof given using complex coordinates...Any ideas?
> >
> > Sincerely,
> >
> > Jeff
• Dear Jeff, The statement with A, B, C and D very much sound to me as P-perpendicular generalization of isogonal conjugacy. All statements for the
May 23, 2005 1 of 6
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Dear Jeff,

The statement with A, B, C and D very much sound to me as "P-perpendicular"
generalization of isogonal conjugacy. All statements for the orthocentric
configuration smoothly generalize to general A,B,C,D by
"P-perpendicularity".

See http://forumgeom.fau.edu/FG2001volume1/FG200122.pdf

Kind regards,
Floor van Lamoen.

> Dear Keith Dean and Floor van Lamoen,
>
> Hope one of you are listening. There is a ball headed in your general
> direction with the word "reciprocal conjugation" or "isoconjugation"
> written on it. I would really appreciate any input you might have
> regarding possible solutions to this problem (especially solutions
> using complex coordinates).
>
> Sincerely,
> Jeff Brooks
>
> PS
> Re: http://forumgeom.fau.edu/FG2001volume1/FG200116.pdf
>
>
>
> > Alexey,
> >
> > Doesn't look like we'll have anyone taking up this proposition
> > anytime soon. How about we limit our observations to orthocentric
> > configurations or maybe generalized circumconics just to get the
> ball
> > rolling.
> >
> > Jeff
> >
> >
> >
> > > Dear Alexey,
> > >
> > > In message 11252, you wrote:
> > >
> > > > Let give 4 points A, B, C, D. X, Y, Z are the common points of
> AB
> > > and CD, AC and BD, AD and BC, P - arbitrary point distinct from
> X,
> > Y,
> > > Z. Then the polars of P wrt all conics passing through A, B, C, D
> > > have the common point P'. If A, B, C, D are orthocentric then P'
> is
> > > isogonally conjugated to P wrt XYZ, and if one of points A, B, C,
> D
> > > is the centroid of three other, then P' is isotomic conjugated.
> > There
> > > is an interesting corollary. Let U, U' and V, V' are two pairs of
> > > conjugated points. Then the points UV^U'V' and U'V^UV' are
> > conjugated.
> > > >
> > >
> > > I believe a direct proof of your statements can be given without
> > > referring to well known geometric theorems. Personally, I would
> > love
> > > to see a proof given using complex coordinates...Any ideas?
> > >
> > > Sincerely,
> > >
> > > Jeff
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