- 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

> - [APH]
> > Let D be a point on the Incircle, D* its antipodal point, and

[FJ]

> > 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?)

> Yes, in this case we have something: all lines m(d) pass through

Dear Francisco

> X(56) = exsimilicenter of (O) and (I).

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 - 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

> - 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

> >

>