QL-L1: The Newton Line

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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.

QL-L1 is called the “mid-diagonal line” by J.W. Clawson in Ref-31.

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:

            l²

Properties:

• QL-P5, QL-P7, QL-P12, QL-P20, QL-P22, QL-P23 are points on QL-L1.
• The 3 QL-versions of QA-P1 lie on QL-L1.
• The 3 QL-versions of QG-P2 lie on QL-L1 with centroid QL-P12. See Ref-66, QPG-message#1271.
• The 3 QL-versions of QG-P12 lie on QL-L1.
• The centers of all inscribed conics of a Quadrilateral lie on QL-L1 (proved by Newton; see also Ref-4, page 49).
• QL-L1 is parallel to QL-L4 (the Morley Line).
• QL-L1 is parallel to the axis of QL-Co1 (the Inscribed Quadrilateral Parabola).
• QL-L1 is perpendicular to QL-L2 (Steiner Line) and QL-L3 (QL-Pedal Line).
• 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 Trilinear Polar of QL-P13 wrt the QL-Diagonal Triangle QL-Tr1 (note Eckart Schmidt).
• QL-L1 is the QA-Orthopole (QA-Tf3) of the line QG-P1.QL-P1.
• One of the 2 asymptotes of QL-Co2 equals QL-L1.
• One of the 3 asymptotes of QL-Cu1 // QL-L1.
• One of the 2 asymptotes of QG-Co3 // QL-L1.
• One of the 3 asymptotes of QA-Cu2 // QL-L1.
• The Involutary Conjugate (QA-Tf2) of QG-P15 constructed in a Quadrigon is the Infinity Point of the Newton Line (QL-L1) in a Quadrigon.
• The axes of the 2 circumscribed QA-parabolas (see QA-2Co1) of the Quadrangle formed by the centroids of the 4 Component Triangles of the Reference Quadrilateral are parallel to QL-L1 and QL-L9 (Eckart Schmidt, September 18, 2012).
• 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.

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