Sunday, August 25, 2013

Parallel and perpendicular central lines



Central lines determined by 8-first ETC centers have many parallel and perpendicular central lines. 

 These results have been found using ETC centers X(1)-X(5543) as published by August 25th, 2013.

 Note: The line at infinity L­∞ is both parallel and perpendicular to any line and has been removed from all lists.
NAGEL LINE (X(1), X(2))=(IG)
  • Parallel lines intersecting L­∞ at X(519):  
  • Perpendicular lines intersecting L­∞ at X(3667):

EULER LINE (X(2), X(3))=(GO)
  • Parallel lines intersecting L­∞ at X(30):
  • Perpendicular lines intersecting L­∞ at X(523):

SODDY LINE (X(1), X(7))=(IGe)
  • Parallel lines intersecting L­∞ at X(516):
  • Perpendicular lines intersecting L­∞ at X(514):

LINE (X(1), X(3))=(IO)
  • Parallel lines intersecting L­∞ at X(517):
  • Perpendicular lines intersecting L­∞ at X(513):

BROCARD AXIS (X(3), X(6))=(OK)
  • Parallel lines intersecting L­∞ at X(511):
  • Perpendicular lines intersecting L­∞ at X(512):

LINE (X(1), X(4))=(IH)
  • Parallel lines intersecting L­∞ at X(515):
  • Perpendicular lines intersecting L­∞ at X(522):

LINE (X(1), X(5))=(IN)
  • Parallel lines intersecting L­∞ at X(952):
  • Perpendicular lines intersecting L­∞ at X(900):

LINE (X(1), X(6))=(IK)
  • Parallel lines intersecting L­∞ at X(518):
  • Perpendicular lines intersecting L­∞ at X(3309):

LINE (X(2), X(6))=(GK)
  • Parallel lines intersecting L­∞ at X(524):
  • Perpendicular lines intersecting L­∞ at X(1499):

LINE (X(2), X(7))=(GGe)
  • Parallel lines intersecting L­∞ at X(527):
  • Perpendicular lines intersecting L­∞ at X(?) (see [1] below):

LINE (X(3), X(7))=(OGe)
  • Parallel lines intersecting L­∞ at X(?) (see [2] below):
  • Perpendicular lines intersecting L­∞ at X(?) (see [3] below):

LINE (X(3), X(8))=(ONa)
  • This line is parallel to line (X(1), X(5) ) = (IN) 

 LINE (X(4), X(6))=(HK)
  • Parallel lines intersecting L­∞ at X(1503):
  • Perpendicular lines intersecting L­∞ at X(525):

LINE (X(4), X(7))=(HGe)
  • Parallel lines intersecting L­∞ at X(971):
  • Perpendicular lines intersecting L­∞ at X(3900):

LINE (X(4), X(8))=(HNa)
  • This line is parallel to line (X(1), X(3))=(IO)

 LINE (X(5), X(6))=(NK)
  • Parallel lines intersecting L­∞ at X(3564):
  • Perpendicular lines intersecting L­∞ at X(3566):

LINE (X(5), X(7))=(NGe)
  • Parallel lines intersecting L­∞ at X(?) (see [4] below):
  • Perpendicular lines intersecting L­∞ at X(?) (see [5] below):

LINE (X(5), X(8))=(NNa)
  • Parallel lines intersecting L­∞ at X(?) (see [6] below):
  • Perpendicular lines intersecting L­∞ at X(?) (see [7] below):

LINE (X(6), X(7))=(KGe)
  • Parallel lines intersecting L­∞ at X(?) (see [8] below):

LINE (X(6)X(8))=(KNa)
  • Parallel lines intersecting L­∞ at X(?) (see [9] below):
  • Perpendicular lines intersecting L­∞ at X(?) (see [10] below):

LINE (X(7), X(8))=(GeNa)
  • This line is parallel to line (X(1), X(6))=(IK)

NOTES Non-ETC intersections:  (All barycentrics)

[1]: (b-c)*(3*a^3-(5*(b+c))*a^2+(b+c)^2*a+(b+c)*(b-c)^2) : :
[2]: 2*a^6-(2*(b+c))*a^5-(3*(b^2+c^2))*a^4+(2*(b+c))*(b^2+c^2)*a^3+(2*(b^3-c^3))*(b-c)*a^2-(b^2-c^2)^2*(b-c)^2 : :

[3]: (b-c)*(2*a^3-(3*(b+c))*a^2+(b^2-c^2)*(b-c)) : :

[4]: 2*a^6+(-7*c^2-7*b^2+4*b*c)*a^4+(2*(b+c))*(b^2+c^2)*a^3+(6*(b^3-c^3))*(b-c)*a^2-2*(b-c)^2*(b+c)^3*a-(b^2-c^2)^2*(b-c)^2 : :

[5]: (b-c)*(4*a^3-(7*(b+c))*a^2+2*(b+c)^2*a+(b^2-c^2)*(b-c)) : :

[6]: 2*a^4-(4*(b+c))*a^3-(c^2+b^2-8*b*c)*a^2+(4*(b^2-c^2))*(b-c)*a-(b^2-c^2)^2 : :

[7]: (b-c)*(b+c-4*a) : :

[8]: 2*a^4-(2*(b+c))*a^3+a^2*(b^2+c^2)-(b^2+c^2)*(b-c)^2 : :

[9]: 2*a^3+(b^2+c^2)*a-(b+c)*(b^2+c^2) : :

[10]: (b-c)*(2*a^3+(b+c)*a^2+(2*(b^2+c^2))*a-(b+c)*(b^2+c^2)) : :
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