Newton line
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]
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, any interior point P on the Newton line of a quadrilateral ABCD has the property that the sum of the areas of the triangles APD and BPC equals the sum of the areas of the triangles APB and CPD.
If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line.[2]
See also
References
- 1 2 Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, pp. 108-109 (online copy, p. 108, at Google Books)
- ↑ Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, The IMO Compendium, Springer, 2006, p. 15.
External links
- Alexander Bogomolny: Bimedians in a Quadrilateral at cut-the-knot.org
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