QL-Tf9: QL- Involutary Centerline at 5th Line touchpoint


Let L be a random line and let Cox be the QL-inscribed conic tangent at L.

The point of tangency where conic and L touch is called the 5th Line touchpoint of L (see QL-Tf7).
QL-Tf9 is the Involutary Centerline (see QL-Tf8) of P = QL-Tf8(P).
QL-Tf9 in a Quadrilateral is the equivalent of QA-Tf5 in a Quadrangle.
 
QL Tf9 Involutary Centerline of the 5th Line point of tangency
 
CT-Coordinates
Let L = (x:y:z), then QL-Tf10(L)=
(m n x (-l m2 n x2 - l m n2 x2 + 3 m2 n2 x2 + l2 m n x y + l2 n2 x y - 3 l m n2 x y + l2 m2 x z + l2 m n x z - 3 l m2 n x z - 2 l3 m y z - 2 l3 n y z + 2 l2 m n y z) :
  l n y (l m2 n x y - 3 l m n2 x y + m2 n2 x y - l2 m n y2 + 3 l2 n2 y2 - l m n2 y2 - 2 l m3 x z + 2 l m2 n x z - 2 m3 n x z + l2 m2 y z - 3 l2 m n y z + l m2 n y z) :
  l m z (2 l m n2 x y - 2 l n3 x y - 2 m n3 x y - 3 l m2 n x z + l m n2 x z + m2 n2 x z - 3 l2 m n y z + l2 n2 y z + l m n2 y z + 3 l2 m2 z2 - l2 m n z2 - l m2 n z2))
 
Properties
QL-Tf9(QL-L1) is a line parallel to QL-L1 having the same distance to QL-L1 as QL-P13, only being positioned at the other side.