Newton line

Short description: Line between midpoints of 2 diagonals in a 4-sided shape with at most 2 parallel sides

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.

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.

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

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

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

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

See also

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

External links

• Weisstein, Eric W.. "Léon Anne's Theorem". http://mathworld.wolfram.com/LeonAnnesTheorem.html. 
• Alexander Bogomolny: Bimedians in a Quadrilateral at cut-the-knot.org