Newton line

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E, K, F lie on a common line, the Newton line Newton line.svg
E, K, F lie on a common line, the Newton line

In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides. [1]

Contents

Properties

The line segments GH and IJ that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line. This point K bisects the line segment EF that connects the diagonal midpoints. [1]

By Anne's theorem and its converse, any interior point P on the Newton line of a quadrilateral ABCD has the property that

where [△ABP] denotes the area of triangle ABP. [2]

If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. [3]

See also

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References

  1. 1 2 Alsina, Claudi; Nelsen, Roger B. (2010). Charming Proofs: A Journey Into Elegant Mathematics. Mathematics Association of America. pp. 108–109. ISBN   9780883853481.
  2. Alsina & Nelsen (2010), pp.  116–117.
  3. Djukić, Dušan; Janković, Vladimir; Matić, Ivan; Petrović, Nikola (2006). The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004. Springer. p. 15. doi:10.1007/0-387-33430-0.