## Parallelogic triangles [corrected]

Expand Messages
• Let T = ABC be a triangle and L a line intersecting the sidelines BC,CA,AB of ABC at A , B , C , resp. Description of a transformation f of T in f(T): Let f(a)
Message 1 of 4 , Jan 25, 2013
Let T = ABC be a triangle and L a line intersecting
the sidelines BC,CA,AB of ABC at A', B', C', resp.

Description of a transformation f of T in f(T):

Let f(a) be the reflection of the perpendicular to BC at A'
in the line L, and similarly f(b), f(c).

Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).

The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.

We have ff(T) = T and f(x) = y and f(y) = x
[==> ff(x) = f(y) = x and ff(y) = f(x) = y]

Which are the coordinates of x,y ?

APH
• If L is the trilinear polar of P=(x:y:z) and X=parallelogic center center of T with respect f(T) Y=parallelogic center center of T with respect f(T) we have
Message 2 of 4 , Jan 25, 2013
If L is the trilinear polar of P=(x:y:z) and
X=parallelogic center center of T with respect f(T)
Y=parallelogic center center of T with respect f(T)

we have known points for X:

for P=X1, X=X953,
for P=X4, X=X477
for P=X6, X=X2698
for P=X7, X=X2724
for P=X8, X=X2734

Coordinates of isotomic conjugate of X and coordinates of Y are below.

Best regards,

Francisco Javier.

----------------

Coordinates of isotomic conjugate of X are

{a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}

Coordinates of Y are

{(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}

--- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
>
> Let T = ABC be a triangle and L a line intersecting
> the sidelines BC,CA,AB of ABC at A', B', C', resp.
>
> Description of a transformation f of T in f(T):
>
> Let f(a) be the reflection of the perpendicular to BC at A'
> in the line L, and similarly f(b), f(c).
>
> Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
>
> The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.
>
> We have ff(T) = T and f(x) = y and f(y) = x
> [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
>
> Which are the coordinates of x,y ?
>
> APH
>
• Sorry, I meant Y = parallelogic center center of f(T) with respect T
Message 3 of 4 , Jan 25, 2013
Sorry, I meant

Y = parallelogic center center of f(T) with respect T

--- In Hyacinthos@yahoogroups.com, "Francisco Javier" wrote:
>
> If L is the trilinear polar of P=(x:y:z) and
> X=parallelogic center center of T with respect f(T)
> Y=parallelogic center center of T with respect f(T)
>
> we have known points for X:
>
> for P=X1, X=X953,
> for P=X4, X=X477
> for P=X6, X=X2698
> for P=X7, X=X2724
> for P=X8, X=X2734
>
> Coordinates of isotomic conjugate of X and coordinates of Y are below.
>
> Best regards,
>
> Francisco Javier.
>
> ----------------
>
> Coordinates of isotomic conjugate of X are
>
> {a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
> 4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
> 2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
> 2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
> a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
> 2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
> b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
> 2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
> a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
> b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
> 2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
> a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
> a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
> 2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
> 2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
> b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
> a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}
>
> Coordinates of Y are
>
> {(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
> b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
> c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
> b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
> 3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
> 2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
> b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
> 3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
> 2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
> 8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
> 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
> b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
> c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
> b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
> b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
> a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
> 6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
> 3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
> a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
> a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
> b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
> b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
> 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
> 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
> b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
> 5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
> 4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
> b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
> 5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
> 4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
> a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
> z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
> c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
> b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
> c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
> 5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
> c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
> 3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
> 6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
> 7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
> 3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
> a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
> a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
> a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
> 4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
> 5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
> a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
> 20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
> a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
> a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
> 7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
> 3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
> c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
> a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
> a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
> 2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
> 4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
> 2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
> 4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
> 4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
> a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
> y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
> c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
> 5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
> c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
> b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
> c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
> b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
> 5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
> a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
> 20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
> a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
> 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
> 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
> b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
> a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
> 3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
> b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
> c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
> b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
> 3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
> c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
> 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
> 2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
> 2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
> a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
> 3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
> 3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
> a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
> 2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}
>
> --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
> >
> > Let T = ABC be a triangle and L a line intersecting
> > the sidelines BC,CA,AB of ABC at A', B', C', resp.
> >
> > Description of a transformation f of T in f(T):
> >
> > Let f(a) be the reflection of the perpendicular to BC at A'
> > in the line L, and similarly f(b), f(c).
> >
> > Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
> >
> > The triangles T, f(T) are parallelogic and let x,y be the parallelogic centers.
> >
> > We have ff(T) = T and f(x) = y and f(y) = x
> > [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
> >
> > Which are the coordinates of x,y ?
> >
> > APH
> >
>
• The points X, Y are on the circumcircles of T, f(T), resp. Right? We can prove or disprove that X is on the circumcircle of T = ABC by this way: Let p(a),
Message 4 of 4 , Jan 26, 2013
The points X, Y are on the circumcircles of T, f(T), resp. Right?

We can prove or disprove that X is on the circumcircle of T = ABC by this
way:

Let p(a), p(b), p(c) be the parallels through A,B,C, to f(a), f(b), f(c),
resp.
and let r(a), r(b), r(c) their respective reflections on the angle
bisectors of T .

If the angles (L, r(a)), (L, r(b)), (L, r(c)) are equal, then the lines
r(a), r(b), r(c)
are parallel and therefore the isogonal conjugate of their point of
concurrence (on the line at infinity)
is lying on the circumcircle of ABC.

APH

2013/1/26 Francisco Javier <garciacapitan@...>

> **
>
>
> If L is the trilinear polar of P=(x:y:z) and
> X=parallelogic center center of T with respect f(T)
> Y = parallelogic center center of f(T) with respect T
>
> we have known points for X:
>
> for P=X1, X=X953,
> for P=X4, X=X477
> for P=X6, X=X2698
> for P=X7, X=X2724
> for P=X8, X=X2734
>
> Coordinates of isotomic conjugate of X and coordinates of Y are below.
>
> Best regards,
>
> Francisco Javier.
>
> ----------------
>
> Coordinates of isotomic conjugate of X are
>
> {a^2 c^2 x^2 y^2 - b^2 c^2 x^2 y^2 - c^4 x^2 y^2 +
> 4 b^2 c^2 x^2 y z - 2 a^2 c^2 x y^2 z - 2 b^2 c^2 x y^2 z +
> 2 c^4 x y^2 z + a^2 b^2 x^2 z^2 - b^4 x^2 z^2 - b^2 c^2 x^2 z^2 -
> 2 a^2 b^2 x y z^2 + 2 b^4 x y z^2 - 2 b^2 c^2 x y z^2 +
> a^2 b^2 y^2 z^2 - b^4 y^2 z^2 + a^2 c^2 y^2 z^2 +
> 2 b^2 c^2 y^2 z^2 - c^4 y^2 z^2, -a^2 c^2 x^2 y^2 +
> b^2 c^2 x^2 y^2 - c^4 x^2 y^2 - 2 a^2 c^2 x^2 y z -
> 2 b^2 c^2 x^2 y z + 2 c^4 x^2 y z + 4 a^2 c^2 x y^2 z -
> a^4 x^2 z^2 + a^2 b^2 x^2 z^2 + 2 a^2 c^2 x^2 z^2 +
> b^2 c^2 x^2 z^2 - c^4 x^2 z^2 + 2 a^4 x y z^2 - 2 a^2 b^2 x y z^2 -
> 2 a^2 c^2 x y z^2 - a^4 y^2 z^2 + a^2 b^2 y^2 z^2 -
> a^2 c^2 y^2 z^2, -a^4 x^2 y^2 + 2 a^2 b^2 x^2 y^2 - b^4 x^2 y^2 +
> a^2 c^2 x^2 y^2 + b^2 c^2 x^2 y^2 - 2 a^2 b^2 x^2 y z +
> 2 b^4 x^2 y z - 2 b^2 c^2 x^2 y z + 2 a^4 x y^2 z -
> 2 a^2 b^2 x y^2 z - 2 a^2 c^2 x y^2 z - a^2 b^2 x^2 z^2 -
> b^4 x^2 z^2 + b^2 c^2 x^2 z^2 + 4 a^2 b^2 x y z^2 - a^4 y^2 z^2 -
> a^2 b^2 y^2 z^2 + a^2 c^2 y^2 z^2}
>
> Coordinates of Y are
>
> {(y - z) (-a^4 c^2 x^4 y^2 + 2 a^2 b^2 c^2 x^4 y^2 -
> b^4 c^2 x^4 y^2 + 2 a^2 c^4 x^4 y^2 + 2 b^2 c^4 x^4 y^2 -
> c^6 x^4 y^2 - a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 -
> b^4 c^2 x^3 y^3 + c^6 x^3 y^3 - a^6 x^4 y z + 3 a^4 b^2 x^4 y z -
> 3 a^2 b^4 x^4 y z + b^6 x^4 y z + 3 a^4 c^2 x^4 y z -
> 2 a^2 b^2 c^2 x^4 y z - b^4 c^2 x^4 y z - 3 a^2 c^4 x^4 y z -
> b^2 c^4 x^4 y z + c^6 x^4 y z + a^6 x^3 y^2 z -
> 3 a^4 b^2 x^3 y^2 z + 3 a^2 b^4 x^3 y^2 z - b^6 x^3 y^2 z -
> 2 a^4 c^2 x^3 y^2 z - 6 a^2 b^2 c^2 x^3 y^2 z +
> 8 b^4 c^2 x^3 y^2 z + a^2 c^4 x^3 y^2 z - 7 b^2 c^4 x^3 y^2 z +
> 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
> b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
> c^6 x^2 y^3 z - a^4 b^2 x^4 z^2 + 2 a^2 b^4 x^4 z^2 -
> b^6 x^4 z^2 + 2 a^2 b^2 c^2 x^4 z^2 + 2 b^4 c^2 x^4 z^2 -
> b^2 c^4 x^4 z^2 + a^6 x^3 y z^2 - 2 a^4 b^2 x^3 y z^2 +
> a^2 b^4 x^3 y z^2 - 3 a^4 c^2 x^3 y z^2 -
> 6 a^2 b^2 c^2 x^3 y z^2 - 7 b^4 c^2 x^3 y z^2 +
> 3 a^2 c^4 x^3 y z^2 + 8 b^2 c^4 x^3 y z^2 - c^6 x^3 y z^2 +
> a^4 b^2 x^2 y^2 z^2 - 2 a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 +
> a^4 c^2 x^2 y^2 z^2 + 20 a^2 b^2 c^2 x^2 y^2 z^2 -
> b^4 c^2 x^2 y^2 z^2 - 2 a^2 c^4 x^2 y^2 z^2 -
> b^2 c^4 x^2 y^2 z^2 + c^6 x^2 y^2 z^2 - a^6 x y^3 z^2 +
> 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
> 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
> b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 +
> 5 a^4 b^2 x^2 y z^3 - 4 a^2 b^4 x^2 y z^3 - b^6 x^2 y z^3 -
> 4 a^2 b^2 c^2 x^2 y z^3 + 2 b^4 c^2 x^2 y z^3 -
> b^2 c^4 x^2 y z^3 - a^6 x y^2 z^3 - 4 a^4 b^2 x y^2 z^3 +
> 5 a^2 b^4 x y^2 z^3 + 2 a^4 c^2 x y^2 z^3 -
> 4 a^2 b^2 c^2 x y^2 z^3 - a^2 c^4 x y^2 z^3 + a^6 y^3 z^3 -
> a^2 b^4 y^3 z^3 + 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3), (x -
> z) (a^4 c^2 x^3 y^3 - 2 a^2 b^2 c^2 x^3 y^3 + b^4 c^2 x^3 y^3 -
> c^6 x^3 y^3 + a^4 c^2 x^2 y^4 - 2 a^2 b^2 c^2 x^2 y^4 +
> b^4 c^2 x^2 y^4 - 2 a^2 c^4 x^2 y^4 - 2 b^2 c^4 x^2 y^4 +
> c^6 x^2 y^4 + a^4 c^2 x^3 y^2 z + 4 a^2 b^2 c^2 x^3 y^2 z -
> 5 b^4 c^2 x^3 y^2 z - 2 a^2 c^4 x^3 y^2 z + 4 b^2 c^4 x^3 y^2 z +
> c^6 x^3 y^2 z + a^6 x^2 y^3 z - 3 a^4 b^2 x^2 y^3 z +
> 3 a^2 b^4 x^2 y^3 z - b^6 x^2 y^3 z - 8 a^4 c^2 x^2 y^3 z +
> 6 a^2 b^2 c^2 x^2 y^3 z + 2 b^4 c^2 x^2 y^3 z +
> 7 a^2 c^4 x^2 y^3 z - b^2 c^4 x^2 y^3 z - a^6 x y^4 z +
> 3 a^4 b^2 x y^4 z - 3 a^2 b^4 x y^4 z + b^6 x y^4 z +
> a^4 c^2 x y^4 z + 2 a^2 b^2 c^2 x y^4 z - 3 b^4 c^2 x y^4 z +
> a^2 c^4 x y^4 z + 3 b^2 c^4 x y^4 z - c^6 x y^4 z +
> a^4 b^2 x^3 y z^2 - 2 a^2 b^4 x^3 y z^2 + b^6 x^3 y z^2 +
> 4 a^2 b^2 c^2 x^3 y z^2 + 4 b^4 c^2 x^3 y z^2 -
> 5 b^2 c^4 x^3 y z^2 - a^6 x^2 y^2 z^2 + 2 a^4 b^2 x^2 y^2 z^2 -
> a^2 b^4 x^2 y^2 z^2 + a^4 c^2 x^2 y^2 z^2 -
> 20 a^2 b^2 c^2 x^2 y^2 z^2 - b^4 c^2 x^2 y^2 z^2 +
> a^2 c^4 x^2 y^2 z^2 + 2 b^2 c^4 x^2 y^2 z^2 - c^6 x^2 y^2 z^2 -
> a^4 b^2 x y^3 z^2 + 2 a^2 b^4 x y^3 z^2 - b^6 x y^3 z^2 +
> 7 a^4 c^2 x y^3 z^2 + 6 a^2 b^2 c^2 x y^3 z^2 +
> 3 b^4 c^2 x y^3 z^2 - 8 a^2 c^4 x y^3 z^2 - 3 b^2 c^4 x y^3 z^2 +
> c^6 x y^3 z^2 + a^6 y^4 z^2 - 2 a^4 b^2 y^4 z^2 +
> a^2 b^4 y^4 z^2 - 2 a^4 c^2 y^4 z^2 - 2 a^2 b^2 c^2 y^4 z^2 +
> a^2 c^4 y^4 z^2 + a^4 b^2 x^3 z^3 - b^6 x^3 z^3 -
> 2 a^2 b^2 c^2 x^3 z^3 + b^2 c^4 x^3 z^3 - 5 a^4 b^2 x^2 y z^3 +
> 4 a^2 b^4 x^2 y z^3 + b^6 x^2 y z^3 + 4 a^2 b^2 c^2 x^2 y z^3 -
> 2 b^4 c^2 x^2 y z^3 + b^2 c^4 x^2 y z^3 + a^6 x y^2 z^3 +
> 4 a^4 b^2 x y^2 z^3 - 5 a^2 b^4 x y^2 z^3 - 2 a^4 c^2 x y^2 z^3 +
> 4 a^2 b^2 c^2 x y^2 z^3 + a^2 c^4 x y^2 z^3 - a^6 y^3 z^3 +
> a^2 b^4 y^3 z^3 - 2 a^2 b^2 c^2 y^3 z^3 + a^2 c^4 y^3 z^3), (x -
> y) (-a^4 c^2 x^3 y^3 + 2 a^2 b^2 c^2 x^3 y^3 - b^4 c^2 x^3 y^3 +
> c^6 x^3 y^3 - a^4 c^2 x^3 y^2 z - 4 a^2 b^2 c^2 x^3 y^2 z +
> 5 b^4 c^2 x^3 y^2 z + 2 a^2 c^4 x^3 y^2 z - 4 b^2 c^4 x^3 y^2 z -
> c^6 x^3 y^2 z + 5 a^4 c^2 x^2 y^3 z - 4 a^2 b^2 c^2 x^2 y^3 z -
> b^4 c^2 x^2 y^3 z - 4 a^2 c^4 x^2 y^3 z + 2 b^2 c^4 x^2 y^3 z -
> c^6 x^2 y^3 z - a^4 b^2 x^3 y z^2 + 2 a^2 b^4 x^3 y z^2 -
> b^6 x^3 y z^2 - 4 a^2 b^2 c^2 x^3 y z^2 - 4 b^4 c^2 x^3 y z^2 +
> 5 b^2 c^4 x^3 y z^2 + a^6 x^2 y^2 z^2 - a^4 b^2 x^2 y^2 z^2 -
> a^2 b^4 x^2 y^2 z^2 + b^6 x^2 y^2 z^2 - 2 a^4 c^2 x^2 y^2 z^2 +
> 20 a^2 b^2 c^2 x^2 y^2 z^2 - 2 b^4 c^2 x^2 y^2 z^2 +
> a^2 c^4 x^2 y^2 z^2 + b^2 c^4 x^2 y^2 z^2 - a^6 x y^3 z^2 +
> 2 a^4 b^2 x y^3 z^2 - a^2 b^4 x y^3 z^2 - 4 a^4 c^2 x y^3 z^2 -
> 4 a^2 b^2 c^2 x y^3 z^2 + 5 a^2 c^4 x y^3 z^2 - a^4 b^2 x^3 z^3 +
> b^6 x^3 z^3 + 2 a^2 b^2 c^2 x^3 z^3 - b^2 c^4 x^3 z^3 -
> a^6 x^2 y z^3 + 8 a^4 b^2 x^2 y z^3 - 7 a^2 b^4 x^2 y z^3 +
> 3 a^4 c^2 x^2 y z^3 - 6 a^2 b^2 c^2 x^2 y z^3 +
> b^4 c^2 x^2 y z^3 - 3 a^2 c^4 x^2 y z^3 - 2 b^2 c^4 x^2 y z^3 +
> c^6 x^2 y z^3 - 7 a^4 b^2 x y^2 z^3 + 8 a^2 b^4 x y^2 z^3 -
> b^6 x y^2 z^3 + a^4 c^2 x y^2 z^3 - 6 a^2 b^2 c^2 x y^2 z^3 +
> 3 b^4 c^2 x y^2 z^3 - 2 a^2 c^4 x y^2 z^3 - 3 b^2 c^4 x y^2 z^3 +
> c^6 x y^2 z^3 + a^6 y^3 z^3 - a^2 b^4 y^3 z^3 +
> 2 a^2 b^2 c^2 y^3 z^3 - a^2 c^4 y^3 z^3 - a^4 b^2 x^2 z^4 +
> 2 a^2 b^4 x^2 z^4 - b^6 x^2 z^4 + 2 a^2 b^2 c^2 x^2 z^4 +
> 2 b^4 c^2 x^2 z^4 - b^2 c^4 x^2 z^4 + a^6 x y z^4 -
> a^4 b^2 x y z^4 - a^2 b^4 x y z^4 + b^6 x y z^4 -
> 3 a^4 c^2 x y z^4 - 2 a^2 b^2 c^2 x y z^4 - 3 b^4 c^2 x y z^4 +
> 3 a^2 c^4 x y z^4 + 3 b^2 c^4 x y z^4 - c^6 x y z^4 -
> a^6 y^2 z^4 + 2 a^4 b^2 y^2 z^4 - a^2 b^4 y^2 z^4 +
> 2 a^4 c^2 y^2 z^4 + 2 a^2 b^2 c^2 y^2 z^4 - a^2 c^4 y^2 z^4)}
>
>
> --- In Hyacinthos@yahoogroups.com, "Antreas" wrote:
> >
> > Let T = ABC be a triangle and L a line intersecting
> > the sidelines BC,CA,AB of ABC at A', B', C', resp.
> >
> > Description of a transformation f of T in f(T):
> >
> > Let f(a) be the reflection of the perpendicular to BC at A'
> > in the line L, and similarly f(b), f(c).
> >
> > Denote: f(T) = the triangle bounded by the lines f(a), f(b), f(c).
> >
> > The triangles T, f(T) are parallelogic and let x,y be the parallelogic
> centers.
> >
> > We have ff(T) = T and f(x) = y and f(y) = x
> > [==> ff(x) = f(y) = x and ff(y) = f(x) = y]
> >
> > Which are the coordinates of x,y ?
> >
> > APH
> >
>
> _
>

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