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PgP Circumconic Configuration

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  • Quang Tuan Bui
    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
    Message 1 of 4 , Nov 2, 2006
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      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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    • Steve Sigur
      ... 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
      Message 2 of 4 , Nov 2, 2006
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        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





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      • Quang Tuan Bui
        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
        Message 3 of 4 , Nov 2, 2006
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          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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        • Steve Sigur
          ... very nice result, and surprising. ... also nice. F to X to Y is a normal mapping of points from conic to conic, usually a projective transformation and not
          Message 4 of 4 , Nov 3, 2006
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            >
            > 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.

            very nice result, and surprising.

            >
            > 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).

            also nice. F to X to Y is a normal mapping of points from conic to
            conic, usually a projective transformation and not usually
            implemented by a circle.

            >
            > 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.

            This particular property is a general property of circles and their
            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.


            >
            > - 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.
            >

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









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