Loading ...
Sorry, an error occurred while loading the content.

Re: Feuerbach and Miquel

Expand Messages
  • garciacapitan
    Yes, in this case we have something: all lines m(d) pass through X(56) = exsimilicenter of (O) and (I). Francisco Javier.
    Message 1 of 9 , Mar 10, 2009
    • 0 Attachment
      Yes, in this case we have something: all lines m(d) pass through X(56) = exsimilicenter of (O) and (I).

      Francisco Javier.


      --- In Hyacinthos@yahoogroups.com, "xpolakis" <xpolakis@...> wrote:
      >
      >
      > [APH]
      > > > > > Let ABC be a triangle, F the Feuerbach point, F* the antipodal
      > > > > > of F on the incircle, and f,f* the tangents to incircle at > > > > > F,F*, respectively.
      > > > > >
      > > > > > Which points (on the circumcircle of ABC) are the Miquel > > > > > > points of the complete quadrilaterals (AB,BC,CA, f) and
      > > > > > (AB,BC,CA,f*)?
      >
      > [FJ]
      > > > > These are X(106) band X(109), respectively.
      >
      > [APH]
      > > > Let's call this line (ie X(106)X(109)) as Feuerbach-Miquel
      > > > line of the incircle (I).
      > > > I am wondering if the similar Feuerbach - Miquel lines
      > > > of the Excircles (Ia),(Ib),(Ic) have some interesting properties.
      > > > (For exampe: Are they concurrent? Bound a triangle
      > > > in perspective with ABC or some other simple triangle?)
      >
      > [FJ]
      > > Dear Antreas, these three lines are not concurrent, nor bound
      > > a triangle perspective with ABC.
      >
      > Dear Francisco
      >
      > Hmmmmm... It seems 3G [= Great God of Geometry] did not make
      > everything perfect in the Euclidean Space :-)
      >
      > I hope that the following was at least made correctly :-)
      >
      > Let D be a point on the Incircle, D* its antipodal point, and
      > d,d* the tangents to Incircle at D,D*, resp.
      >
      > Let m(d) be the line joining the Miquel points (on the
      > circumcircle) of the complete quadrilaterals (AB,BC,CA, d)
      > and (AB,BC,CA, d*).
      >
      > As D moves on the Incircle, which is the envelope of the line m(d)?
      > Is it a zero circle? (ie are they concurrent? If yes, at which
      > point?)
      >
      > APH
      >
    • xpolakis
      [APH] ... [FJ] ... Dear Francisco I think that the general is true: Let p be a line (trilinear polar of a point P = (x:y:z)) and Q a point on the line. Let
      Message 2 of 9 , Mar 11, 2009
      • 0 Attachment
        [APH]
        > > Let D be a point on the Incircle, D* its antipodal point, and
        > > d,d* the tangents to Incircle at D,D*, resp.
        > >
        > > Let m(d) be the line joining the Miquel points (on the
        > > circumcircle) of the complete quadrilaterals (AB,BC,CA, d)
        > > and (AB,BC,CA, d*).
        > >
        > > As D moves on the Incircle, which is the envelope of the
        > > line m(d)?
        > > Is it a zero circle? (ie are they concurrent? If yes, at which
        > > point?)

        [FJ]
        > Yes, in this case we have something: all lines m(d) pass through
        > X(56) = exsimilicenter of (O) and (I).

        Dear Francisco

        I think that the general is true:

        Let p be a line (trilinear polar of a point P = (x:y:z)) and Q
        a point on the line.
        Let q,q* be the tangents to incircle from Q, and m(q) the
        line joining the Miquel points of the complete
        4laterals (AB,BC,CA,q) and (AB,BC,CA, q*).
        As Q moves on the line p, the lines m(q) concur at some point.
        [In the case of the tangents at antipodal points of the incircle,
        the line p is the Linf]

        Another problem is:
        If Q moves on some circle (instead of line), which is the envelope
        of m(q)? Special Case: Q moves on the circumcircle.

        APH
      • garciacapitan
        Dear Antreas: here we have the following mapping: If Q=(u:v:w), then the line m(q) is (s-a)((s-a)u - a (v+w)) x / a^2 + (s-b)((s-b)v - b (w+u)) y / b^2 +
        Message 3 of 9 , Mar 11, 2009
        • 0 Attachment
          Dear Antreas: here we have the following mapping:

          If Q=(u:v:w), then the line m(q) is

          (s-a)((s-a)u - a (v+w)) x / a^2
          + (s-b)((s-b)v - b (w+u)) y / b^2
          + (s-c)((s-c)w - c (u+v)) z / c^2 = 0

          Hence, if Q is on the line px + qy + rz = 0 then all lines m(q) pass through the point

          (a^2((s - a) p - b q - c r)/(s - a) :
          b^2((s - b) q - c r - a p)/(s - b) :
          c^2((s - c) r - a p - b q)/(s - c)).




          --- In Hyacinthos@yahoogroups.com, "xpolakis" <xpolakis@...> wrote:
          >
          > [APH]
          > > > Let D be a point on the Incircle, D* its antipodal point, and
          > > > d,d* the tangents to Incircle at D,D*, resp.
          > > >
          > > > Let m(d) be the line joining the Miquel points (on the
          > > > circumcircle) of the complete quadrilaterals (AB,BC,CA, d)
          > > > and (AB,BC,CA, d*).
          > > >
          > > > As D moves on the Incircle, which is the envelope of the
          > > > line m(d)?
          > > > Is it a zero circle? (ie are they concurrent? If yes, at which
          > > > point?)
          >
          > [FJ]
          > > Yes, in this case we have something: all lines m(d) pass through
          > > X(56) = exsimilicenter of (O) and (I).
          >
          > Dear Francisco
          >
          > I think that the general is true:
          >
          > Let p be a line (trilinear polar of a point P = (x:y:z)) and Q
          > a point on the line.
          > Let q,q* be the tangents to incircle from Q, and m(q) the
          > line joining the Miquel points of the complete
          > 4laterals (AB,BC,CA,q) and (AB,BC,CA, q*).
          > As Q moves on the line p, the lines m(q) concur at some point.
          > [In the case of the tangents at antipodal points of the incircle,
          > the line p is the Linf]
          >
          > Another problem is:
          > If Q moves on some circle (instead of line), which is the envelope
          > of m(q)? Special Case: Q moves on the circumcircle.
          >
          > APH
          >
        • garciacapitan
          And, if Q is on the circumcircle, then the line m(q) evolves an ellipse centererd on the line OI. The center of this ellipse is the point (a (a+b-c) (a-b+c) (4
          Message 4 of 9 , Mar 11, 2009
          • 0 Attachment
            And, if Q is on the circumcircle, then the line m(q) evolves an ellipse centererd on the line OI. The center of this ellipse is the point

            (a (a+b-c) (a-b+c) (4 a^4-5 a^3 b-3 a^2 b^2+5 a b^3-b^4-5 a^3 c+15 a^2 b c-6 a b^2 c-3 a^2 c^2-6 a b c^2+2 b^2 c^2+5 a c^3-c^4) :

            b (a+b-c) (-a+b+c) (-a^4+5 a^3 b-3 a^2 b^2-5 a b^3+4 b^4-6 a^2 b c+15 a b^2 c-5 b^3 c+2 a^2 c^2-6 a b c^2-3 b^2 c^2+5 b c^3-c^4) :

            c (a-b+c) (-a+b+c) (-a^4+2 a^2 b^2-b^4+5 a^3 c-6 a^2 b c-6 a b^2 c+5 b^3 c-3 a^2 c^2+15 a b c^2-3 b^2 c^2-5 a c^3-5 b c^3+4 c^4)).




            --- In Hyacinthos@yahoogroups.com, "garciacapitan" <garciacapitan@...> wrote:
            >
            > Dear Antreas: here we have the following mapping:
            >
            > If Q=(u:v:w), then the line m(q) is
            >
            > (s-a)((s-a)u - a (v+w)) x / a^2
            > + (s-b)((s-b)v - b (w+u)) y / b^2
            > + (s-c)((s-c)w - c (u+v)) z / c^2 = 0
            >
            > Hence, if Q is on the line px + qy + rz = 0 then all lines m(q) pass through the point
            >
            > (a^2((s - a) p - b q - c r)/(s - a) :
            > b^2((s - b) q - c r - a p)/(s - b) :
            > c^2((s - c) r - a p - b q)/(s - c)).
            >
            >
            >
            >
            > --- In Hyacinthos@yahoogroups.com, "xpolakis" <xpolakis@> wrote:
            > >
            > > [APH]
            > > > > Let D be a point on the Incircle, D* its antipodal point, and
            > > > > d,d* the tangents to Incircle at D,D*, resp.
            > > > >
            > > > > Let m(d) be the line joining the Miquel points (on the
            > > > > circumcircle) of the complete quadrilaterals (AB,BC,CA, d)
            > > > > and (AB,BC,CA, d*).
            > > > >
            > > > > As D moves on the Incircle, which is the envelope of the
            > > > > line m(d)?
            > > > > Is it a zero circle? (ie are they concurrent? If yes, at which
            > > > > point?)
            > >
            > > [FJ]
            > > > Yes, in this case we have something: all lines m(d) pass through
            > > > X(56) = exsimilicenter of (O) and (I).
            > >
            > > Dear Francisco
            > >
            > > I think that the general is true:
            > >
            > > Let p be a line (trilinear polar of a point P = (x:y:z)) and Q
            > > a point on the line.
            > > Let q,q* be the tangents to incircle from Q, and m(q) the
            > > line joining the Miquel points of the complete
            > > 4laterals (AB,BC,CA,q) and (AB,BC,CA, q*).
            > > As Q moves on the line p, the lines m(q) concur at some point.
            > > [In the case of the tangents at antipodal points of the incircle,
            > > the line p is the Linf]
            > >
            > > Another problem is:
            > > If Q moves on some circle (instead of line), which is the envelope
            > > of m(q)? Special Case: Q moves on the circumcircle.
            > >
            > > APH
            > >
            >
          Your message has been successfully submitted and would be delivered to recipients shortly.