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]

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

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

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