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In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side.

The three medians divide the triangle into six smaller triangles of equal area.

Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.

By definition, $AD=DB,\; AF=FC,\; BE=EC\; ,$, thus $[ADO]=[BDO],\; [AFO]=[CFO],\; [BEO]=[CEO]\; ,$, where $[ABC]$ represents the area of triangle $triangle\; ABC$.

We have:

- $[ABO]=[ABE]-[BEO]\; ,$

- $[ACO]=[ACE]-[CEO]\; ,$

Thus, $[ABO]=[ACO]\; ,$ and $[ADO]=[DBO],\; [ADO]=frac\{1\}\{2\}[ABO]$

Since $[AFO]=[FCO],\; [AFO]=\; frac\{1\}\{2\}AFO=frac\{1\}\{2\}[ABO]=[ADO]$, therefore, $[AFO]=[FCO]=[DBO]=[ADO],$. Using the same method, you can show that $[AFO]=[FCO]=[DBO]=[ADO]=[BEO]=[CEO]\; ,$.

- $m\; =\; sqrt\; \{frac\{2\; b^2\; +\; 2\; c^2\; -\; a^2\}\{4\}\; \}$

where a is the side of the triangle whose midpoint is the extreme point of median m.

- Medians and Area Bisectors of a Triangle
- The Medians at cut-the-knot
- Area of Median Triangle at cut-the-knot
- Medians of a triangle With interactive animation
- Constructing a median of a triangle with compass and straightedge animated demonstration

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Last updated on Friday October 03, 2008 at 09:22:47 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 03, 2008 at 09:22:47 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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