Newton line

Template:Mvar 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 other than a parallelogram.Template:R

Properties

The line segments Template:Mvar and Template:Mvar 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 Template:Mvar bisects the line segment Template:Mvar that connects the diagonal midpoints.Template:R

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

$[\mathrm{△}ABP]+[\mathrm{△}CDP]=[\mathrm{△}ADP]+[\mathrm{△}BCP],$

where Template:Math denotes the area of triangle Template:Math.Template:Sfnp

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

See also

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

External links

• Template:MathWorld
• Alexander Bogomolny: Bimedians in a Quadrilateral at cut-the-knot.org