Midpoint theorem (triangle): Difference between revisions

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{{Short description|Geometric theorem involving midpoints on a triangle}}

[[File:Midpoint theorem.svg|thumb|upright=1.25|<math> \begin{align} &\text{D and E midpoints of AC and BC}\\ \Rightarrow \, &DE \parallel AB\text{ and } 2|DE|=|AB|\end{align}</math>]]

The '''midpoint theorem''', '''midsegment theorem''', or '''midline theorem''' states that if the midpoints of two sides of a triangle are connected, then the resulting line segment will be parallel to the third side and have half of its length. The midpoint theorem generalizes to the [[intercept theorem]], where rather than using midpoints, both sides are partitioned in the same ratio.<ref>{{Cite book |last=Clapham |first=Christopher |url=https://en.wikipedia.org/wiki/Special:BookSources/9780199235940 |title=The concise Oxford dictionary of mathematics: clear definitions of even the most complex mathematical terms and concepts |last2=Nicholson |first2=James |date=2009 |publisher=Oxford Univ. Press |isbn=978-0-19-923594-0 |edition=4th |series=Oxford paperback reference |___location=Oxford |pages=297}}</ref><ref>{{Cite book |last=French |first=Doug |url=https://en.wikipedia.org/wiki/Special:BookSources/9780826473622 |title=Teaching and learning geometry: issues and methods in mathematical education |date=2004 |publisher=Continuum |isbn=978-0-8264-7362-2 |___location=London; New York |pages=81–84 |oclc=ocm56658329}}</ref>

The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.

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==See also==

* [[Median of the Trapezoidtrapezoid theorem]]

==References==

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==External links ==

*[http://math.fau.edu/yiu/Oldwebsites/MSTHM2014/Supplement0627.pdf ''The midpoint theorem and its converse'']

*[https://www.edumple.com/cbse-class-9/mathmatics/motivate-in-a-parallelogram-the-diagonals-bisect-each-other-and-conversely/notes/saira_2419 The Mid-Point Theorem'']

*[https://www.youtube.com/watch?v=Q457DRC33zY ''Midpoint theorem and converse Euclidean explained Grade 10+12 ''] (video, 5:28 mins)

*[https://proofwiki.org/wiki/Midline_Theorem ''midpoint theorem''] at the Proof Wiki