QL-Tf9: QL- Involutary Centerline at 5^th 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 5^th 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.

CT-Coordinates

Let L = (x:y:z), then QL-Tf9(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.

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