**QL-L1: The Newton Line**

The intersection point of two lines of a quadrilateral and the intersection point of the other two lines of a quadrilateral are called opposite points and their connective line is called a diagonal. The midpoints of the 3 diagonals of a Quadrilateral are collinear. The line through these 3 midpoints is the Newton Line.

Newton discovered this line because all centers of inscribed conics in a Quadrilateral reside on this line. Gauss discovered that this line of Newton’s also passes through the midpoints of the diagonals of the quadrilateral. That’s why the Newton Line is also known as Gauss Line or Gauss-Newton Line.

Jean-Louis Ayme wrote a book with 30 proofs about this line at Ref-3.

Also he wrote Ref-2a about the Newton Line as well as the Steiner Line.

*Coefficients:**1st CT-coefficient:*

l m + l n - m n (note that this formula is independent of a,b,c)

*1st DT-coefficient:*

*l*

^{2}

*Properties:*-
QL-L1 is the locus of homothetic centers of L1.L2.L3.L4 with the 4 lines formed by the parallel lines to L1, L2, L3, L4 through Euler Line points Q1, Q2, Q3, Q4 of the Component Triangles, where Qi have proportional ratio wrt OrthoCenter and CircumCenter on respective Euler Line.
- QL-L1 is the QA-Orthopole (QA-Tf3) of the line QG-P1.QL-P1.
- Let S12, S23, S34, S41 be the intersection points of QL-L1 in a Quadrigon P1.P2.P3.P4. Now P1.S12/P2.S12 = -P2.P23/P3.S23 = P3.S34/P4.S34 = -P4.S41/P1.S41 (Eckart Schmidt, October 5, 2012).
- All sides of the medial triangles of all 4 Component Triangles of L1.L2.L3.L4 pass through one of the 3 midpoints of the 3 QL-diagonals on QL-L1.
- The Orthopole of ANY line parallel to the Newton line QL-L1 wrt ANY Component Triangle of the Reference Quadrilateral lies on QL-L3. See Ref-34, Seiichi Kirikami, QFG message # 1102.