Newton line

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

Not to be confused with Newton-Gauss line or Isaac Newton line.

In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral other than a parallelogram.^[1]

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

${\displaystyle [\triangle ABP]+[\triangle CDP]=[\triangle ADP]+[\triangle BCP],}$

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

• Complete quadrangle
• Newton's theorem (quadrilateral)
• Newton–Gauss line

References

1. 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. Problem Books in Mathematics. Springer. p. 15. doi:10.1007/0-387-33430-0. ISBN 0-387-24299-6.

External links

• Weisstein, Eric W. "Léon Anne's Theorem". MathWorld.
• Alexander Bogomolny: Bimedians in a Quadrilateral at cut-the-knot.org