nL-n-Luc5e: nL-Par1/Par2 constructions

nL-n-Luc5e transforms a point (n-1)L-n-Px into a similar point nL-n-Px by drawing lines through the n versions of (n-1)L-n-Px in a reference n-Line parallel to the omitted line.

• Suppose the reference n-Line is called Ref.
• When drawing parallel lines through the n versions of (n-1)L-n-Px the result will be an n-Line called Par1.
• When drawing another generation (acting like Par1 is Ref) the resulting n-Line will be called Par2.
• When the corresponding lines of Par1 and Par2 are parallel and Par1 and Par2 are perspective, then this Perspective Center will be called Homothetic Center Par1/Par2-HC(nL-n-Px)).

Par1/Par2-Level-up constructions on nL-n-Px
It appears that :
* nL-n-Luc5e(3L-n-Pi) = 4L-n-Pi for i = 1,...,11
* nL-n-Luc5e(nL-n-P8) = (n+1)L-n-P8 for n = 4,5,6,7,8,9, . . .
* nL-n-Luc5e(nL-n-P3) exists for n = 4,5,6,7 (then possible end of homothecy)
* nL-n-Luc5e(nL-n-P5) exists for n = 4,5,6 (then possible end of homothecy)
* nL-n-Luc5e(nL-n-P7) exists for n = 4,5,6,7,8 (9 gives calculation problems)

5L-n-P7. 5L-o-P1 (1:2)
4L-n-P12 =
= QL-P4    = CC(H(2))
5L-n-P12 =
5L-n-P7. 5L-n-P3 (1:1)
No linear relation with known 6L-points.
= QL-P28 = CC(H(3))
5L-n-P13 =
No lin. rel. with known 5L-points.
6L-n-P13 =
No linear relation with known 6L-points.
4L-n-P14 =
= QL-P29 = CC(H(-2))
5L-n-P14 =
5L-n-P7. 5L-n-P5 (2:-1)
6L-n-P14 =
No linear relation with known 6L-points.
    CC(H(i)) = Center of the 4L-Centercircle wrt HofstadterPoint(i).

Par1/Par2 constructions on nL-Hofstadter Points
* In a 5-Line starting with 4L- n-P13 (QL-P28) as Central Point for the Component 4-Lines it appears that 5L-Par1 is homothetic with 5L-Par2 giving a Homothetic Center 5L-n-P13.
* In a 6-Line starting with 5L- n-P13 as Central Point for the Component 5-Lines it appears that 6L-Par1 is homothetic with 6L-Par2 giving a Homothetic Center 6L- n-P13.
* In a 7-Line starting with 6L- n-P13 as Central Point for the Component 5-Lines it appears that 7L-Par1 is homothetic with 7L-Par2 giving a Homothetic Center 7L- n-P13.
* etc.
This process can be repeated for all other known QL-points generated from Hofstadter Points X(3), X(186), X(256), X(5961), X(5962), X(5963), X(5964).
Corresponding Central Points in the 4-Line will be QL-P4 (wrt X(3)), QL-P28 (wrt X(186)), QL-P29 (wrt X(256)).

I checked it graphically in Cabri for X(256) up to level n=6. Further drawings for n>6 were impossible because of the many internal calculations for the drawing software.

So I checked them with Mathematica Software.
Again there were limitations wrt the many internal calculations.
However there were no contra indications for:
X(186) up to level n=8
X(256) up to level n=7
X(5961) up to level n=7
X(5962) up to level n=7
X(5963) up to level n=6
X(5964) up to level n=4

Therefore I feel confident enough for this conjecture :

Let 3L-P(i) be n-Angle Centers P(i) in a Triangle as described in QFG-message #1872, where i <> -1, 0, 1.
Let 4L-Q(i) be the Circumcenter of the 4 versions of 3L-P(i) in a 4-Line.
For these points nL-Par1 will be homothetic with nL-Par2 using (n-1)L-Q(i) as Central Point, producing new Homothetic Center nL-Q(i), for all n > 4.