QL-Tf5:  QL-Orthopolar Line


Let L1,L2,L3,L4 be a quadrilateral and let L be a random line.
Let X1,X2,X3,X4 be the Orthopoles (for definition see Ref-13) of L resp. wrt triangles (L2,L3,L4), (L3,L4,L1), (L4,L1,L2), (L1,L2,L3).
X1, X2, X3, X4 are collinear on a line being called the QL-Orthopolar Line, which is QL-Tf5(L).
It is remarkable that QL-Tf5(L) // QL-L2 for all L.
This line was already described in the beginning of the 20th century. See Ref-13, keyword “Orthopolar Line”. See also Ref-11, Hyacinthos#21070, Ref-34, QFG#2069, #2089.
 
QL Tf5 Orthopolar Line 01
 
1st CT-coefficient of QL-Tf5[L(x:y:z)]:
SA2 (-l m (y - z)2 + l n (y - z)2)
+ SB SC (-m n x (y - z) + l n y (x - z) - l m z (x - y))
+ SA SC (-m n x (y - z) - l m (x - y) (x - y + z) + l n (x2 - x y + y2 - y z))
+ SA SB (-m n x (y - z) + l n (x - z) (x + y - z) + l m (-x2 + x z + y z - z2))
 
Properties:
• The QL-Orthopolar Lines of QL-L1, QL-L4, QL-L5 are identical being the Pedal Line QL-L3. See Ref-34, QFG#2069.
• The QL-Orthopolar Line of QL-L3 is the Steiner Line QL-L2. See Ref-34, QFG#2069.
QL-Tf5(L) is orthogonal to QL-L1 through the intersection of L and a tangent to QL-Co1 perpendicular L. See Ref-34, QFG#2089.
• The QL-Orthopolar Line of tangents at QL-Co1 is QL-L2. See Ref-34, QFG#2089.
• All lines with the same QL-Orthopolar Line L (perpendicular to QL-L1) are tangents of a parabola with the same focus and axis as QL-Co1 intersecting QL-Co1 orthogonal in the intersections with L. See Ref-34, QFG#2089.