Newton line
{{Short description|Line between midpoints of 2 diagonals in a 4-sided shape with at most 2 parallel sides}}
{{distinguish|Newton-Gauss line|Isaac 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.{{r|alsina}}
Properties
The line segments {{mvar|{{overline|GH}}}} and {{mvar|{{overline|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 {{mvar|K}} bisects the line segment {{mvar|{{overline|EF}}}} that connects the diagonal midpoints.{{r|alsina}}
By Anne's theorem and its converse, any interior point P on the Newton line of a quadrilateral {{mvar|ABCD}} has the property that
:
where {{math|[△{{mvar|ABP}}]}} denotes the area of triangle {{math|△{{mvar|ABP}}}}.{{sfnp|Alsina|Nelsen|2010|pp=[https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA116 116–117]}}
If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line.{{r|vic}}
See also
References
{{reflist|refs =
| last1 = Alsina | first1 = Claudi
| last2 = Nelsen | first2 = Roger B.
| year = 2010
| title = Charming Proofs: A Journey Into Elegant Mathematics
| publisher = Mathematics Association of America
| isbn = 9780883853481
| pages = 108–109
| url = https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA108
}}
| last1 = Djukić | first1 = Dušan
| last2 = Janković | first2 = Vladimir
| last3 = Matić | first3 = Ivan
| last4 = Petrović | first4 = Nikola
| year = 2006
| title = The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2004
| series = Problem Books in Mathematics
| publisher = Springer
| page = 15
| doi = 10.1007/0-387-33430-0
| isbn = 0-387-24299-6
}}
}}
External links
- {{MathWorld |urlname=LeonAnnesTheorem |title=Léon Anne's Theorem}}
- Alexander Bogomolny: [http://www.cut-the-knot.org/Curriculum/Geometry/AnyQuadri.shtml#explanation Bimedians in a Quadrilateral] at cut-the-knot.org