- Dear All My Friends,

There is one special type of circumconic: the circumconic passing through one point P and its isogonal conjugate gP (I still don't know if its special name exists). This PgP circumconic (or shortcut: PgP conic) has some special features which I observe as following.

Let F is fourth intersection of PgP conic with circumcircle (O) of reference triangle ABC.

The results:

1. Any circle (Ox) passing through P, gP will cut circumcircle (O) if and only if it cuts PgP conic at third point X other than P and gP.

All following results are for this ''cutting case''.

2. Let fourth intersection point of PgP conic with circle (Ox) is X' (other than X, P, gP). (In special case X' can be X). Let Y and Y' are intersection points of circle (Ox) with circumcircle (O) so one of them, say Y, is collinear with X, F; the other one, Y', is collinear with X', F. (In special case Y' can be Y).

3. There are two special positions of X on PgP conic such that the circle (Ox) touches circumcircle (O) and touches PgP conic (in this cases: X=X' and Y=Y'). These two positions can be constructed by compass and ruler as following:

- The lines YY' always cuts line PgP at one fixed point, say Q, so Q can be constructed.

- Construct the circle taken QO as diameter. This circle cuts circumcircle (O) at two points, say R, S.

- Construct the circumcircle (Or) of RPgP and circumcircle (Os) of SPgP

- Other than R, the line FR cuts circle (Or) at Xr. Other than S, the line FS cuts circle (Os) at Xs.

- Xr and Xs are two special positions on PgP conic: the circle (Or) touches circumcircle (O) at R and touches PgP conic at Xr; the circle (Os) touches circumcircle (O) at S and touches PgP conic at Xs.

4. Two common tangent lines at Xr, Xs are concurrent with line XX' at infinite point, say Qi. Two common tangent lines at R, S are concurrent with line PgP at mentioned point Q.

5. Midpoint of XrXs is center of PgP conic.

By my opinion: a lot of P in ETC follows this configuration.

Please kindly give me advices, references, calculation coordinates or synthetic proofs!

Thank you and best regards,

Bui Quang Tuan

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[Non-text portions of this message have been removed] - On Nov 2, 2006, at 3:59 AM, Quang Tuan Bui wrote:

> There is one special type of circumconic: the circumconic passing

For years I have been calling this a "Mineur conic" after Adolph

> through one point P and its isogonal conjugate gP (I still don't

> know if its special name exists)

Mineur who studied it a long time ago. In addition to normal triangle

symmetries, it has a symmetry under conjugation, that gives it very

special properties.

The Jerabek circumconic is the most famous example, P and gP being H

and O.

The supplements of P and gP are also on the conic.

You might try the isotomic equivalent using P and tP.

Your results are very nice, by the way.

Steve

http://paideiaschool.org/TeacherPages/Steve_Sigur/interesting2.htm

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[Non-text portions of this message have been removed] - Dear Steve and All My Friends,

Thank you very much for your reference and advice message.

I just also have found why we often see this configuration in ETC. It is because all geometry objects here are combined in one compact configuration by three very power operations:

- isogonal conjugate

- circumconic

- Psi operation (recall that this operation is presented in ETC, before X(98) point)

In fact, in this configuration:

Y = Psi(P, X) = Psi(gP, X)

Y' = Psi(P, X') = Psi(gP, X')

R = Psi(P, Xr) = Psi(gP, Xr)

S = Psi(P, Xs) = Psi(gP, Xs)

In this compact configuration, these operations generate some concyclic poitns. In their order, these concyclic points with circumcircle and circumconic can generate some special features such as concurrencies and tangencies.

Conclusion: what can be worth in this configuration that it can be used as one demonstration of some features of Psi transform. NICE (as Steve kindly give me a word) but NO magical facts here.

Thank you and best regards,

Bui Quang Tuan

Steve Sigur <s.sigur@...> wrote:On Nov 2, 2006, at 3:59 AM, Quang Tuan Bui wrote:

> There is one special type of circumconic: the circumconic passing

> through one point P and its isogonal conjugate gP (I still don't

> know if its special name exists)

For years I have been calling this a "Mineur conic" after Adolph

Mineur who studied it a long time ago. In addition to normal triangle

symmetries, it has a symmetry under conjugation, that gives it very

special properties.

The Jerabek circumconic is the most famous example, P and gP being H

and O.

The supplements of P and gP are also on the conic.

You might try the isotomic equivalent using P and tP.

Your results are very nice, by the way.

Steve

http://paideiaschool.org/TeacherPages/Steve_Sigur/interesting2.htm

The arml website is now officially on line. Find it here.

The Secrets of Georgia ARML 1

Find it here.

The Secrets of Georgia ARML 3 website, is at

http://stuff.mit.edu/~borisa/georgia-arml

Username: arml

Password: soga3

http://www.mandelbrot.org

[Non-text portions of this message have been removed]

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[Non-text portions of this message have been removed] >

very nice result, and surprising.

> Let F is fourth intersection of PgP conic with circumcircle (O) of

> reference triangle ABC.

> The results:

> 1. Any circle (Ox) passing through P, gP will cut circumcircle (O)

> if and only if it cuts PgP conic at third point X other than P and gP.

>

also nice. F to X to Y is a normal mapping of points from conic to

> All following results are for this ''cutting case''.

> 2. Let fourth intersection point of PgP conic with circle (Ox) is

> X' (other than X, P, gP). (In special case X' can be X). Let Y and

> Y' are intersection points of circle (Ox) with circumcircle (O) so

> one of them, say Y, is collinear with X, F; the other one, Y', is

> collinear with X', F. (In special case Y' can be Y).

conic, usually a projective transformation and not usually

implemented by a circle.

>

This particular property is a general property of circles and their

> 3. There are two special positions of X on PgP conic such that the

> circle (Ox) touches circumcircle (O) and touches PgP conic (in this

> cases: X=X' and Y=Y'). These two positions can be constructed by

> compass and ruler as following:

> - The lines YY' always cuts line PgP at one fixed point, say Q, so

> Q can be constructed.

radical axes and not specific to any triangle or any type of conjugacy:

Let l be a fixed line and c a fixed circle. If P and Q are on the

line take the family of circles through P and Q, the radical axes of

all these circles will go through a fixed point on PQ.

>

The points Xr and Xs and the line Xr and Xs are very special. This

> - Construct the circle taken QO as diameter. This circle cuts

> circumcircle (O) at two points, say R, S.

> - Construct the circumcircle (Or) of RPgP and circumcircle (Os) of

> SPgP

> - Other than R, the line FR cuts circle (Or) at Xr. Other than S,

> the line FS cuts circle (Os) at Xs.

> - Xr and Xs are two special positions on PgP conic: the circle (Or)

> touches circumcircle (O) at R and touches PgP conic at Xr; the

> circle (Os) touches circumcircle (O) at S and touches PgP conic at Xs.

> 4. Two common tangent lines at Xr, Xs are concurrent with line XX'

> at infinite point, say Qi. Two common tangent lines at R, S are

> concurrent with line PgP at mentioned point Q.

> 5. Midpoint of XrXs is center of PgP conic.

>

allows the construction of interesting pairwise points for any Mineur

type conic just as the Fermats are constructed for the Kiepert

hyperbola.

I like this construction. It is all very surprising.

Regards,

Steve

[Non-text portions of this message have been removed]