- 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,

> 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,

> Z. Then the polars of P wrt all conics passing through A, B, C, D

There

> 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.

> is an interesting corollary. Let U, U' and V, V' are two pairs of

conjugated.

> conjugated points. Then the points UV^U'V' and U'V^UV' are

> >

love

>

> I believe a direct proof of your statements can be given without

> referring to well known geometric theorems. Personally, I would

> 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 maybe generalized circumconics just to get the ball

rolling.

Jeff

> Dear Alexey,

Y,

>

> 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,

> Z. Then the polars of P wrt all conics passing through A, B, C, D

There

> 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.

> is an interesting corollary. Let U, U' and V, V' are two pairs of

conjugated.

> conjugated points. Then the points UV^U'V' and U'V^UV' are

> >

love

>

> I believe a direct proof of your statements can be given without

> referring to well known geometric theorems. Personally, I would

> 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 "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

I am a long-time admirer of your works,

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

> Alexey,

ball

>

> 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

> rolling.

AB

>

> Jeff

>

>

>

> > Dear Alexey,

> >

> > In message 11252, you wrote:

> >

> > > Let give 4 points A, B, C, D. X, Y, Z are the common points of

> > and CD, AC and BD, AD and BC, P - arbitrary point distinct from

X,

> Y,

is

> > 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'

> > 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 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

> I am a long-time admirer of your works,

> 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