PlanetMath: common point of triangle medians

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common point of triangle medians

(Theorem)

Theorem. The three medians of a triangle intersect one another in one point, which divides each median in the ratio $2\!:\!1$ .

$\begin{pspicture}(-1,-0.5)(5,5.5) \psline(4,0)(3,5) \psdot[linecolor=blue](0,0) ... ...r=red](3,5)(2.33,1.67) \psline[linestyle=dotted](2.33,1.67)(2,0) \end{pspicture}$

Proof. Let the medians of a triangle $ABC$ be $AD$ , $BE$ and $CF$ . Any median vector is the arithmetic mean of the side vectors emanating from the same vertex. Using vectors, let us form three ways all beginning from the vertex $A$ , the first going simply $2/3$ of the median vector $\overrightarrow{AD}$ (blue in the picture):

$\displaystyle \frac{2}{3}\overrightarrow{AD} = \frac{2}{3}\cdot\frac{1}{2}(\ove... ...AB}+\overrightarrow{AC}) = \frac{1}{3}(\overrightarrow{AB}+\overrightarrow{AC})$

(1)

The second way goes first the side vector $\overrightarrow{AB}$ and then $2/3$ of the median vector $\overrightarrow{BE}$ (green in the picture):

$\displaystyle \overrightarrow{AB}+\frac{2}{3}\overrightarrow{BE} = \overrightar... ...errightarrow{AB})\right] = \frac{1}{3}(\overrightarrow{AB}+\overrightarrow{AC})$

(2)

Similarly, the third way goes first the side vector $\overrightarrow{AC}$ and then $2/3$ of the median vector $\overrightarrow{CF}$ (red in the picture):

$\displaystyle \overrightarrow{AC}+\frac{2}{3}\overrightarrow{CF} = \overrightar... ...errightarrow{AC})\right] = \frac{1}{3}(\overrightarrow{AB}+\overrightarrow{AC})$

(3)

Thus the ways (2) and (3), where one goes from $A$ to another vertex and continues along the corresponding median $2/3$ of its length, lead to the point $M$ which is attained directly along $AD$ . This means that all medians intersect in $M$ . The distance of $M$ from any vertex is $2/3$ of the corresponding median, and so the rest of the median is $1/3$ of its length, i.e. the ratio of the parts of any median is $2\!:\!1$ .

"common point of triangle medians" is owned by pahio.

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Cross-references: distance, vectors, vertex, side vectors, arithmetic mean, median vector, proof, ratio, divides, point, intersect, triangle, theorem
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This is version 5 of common point of triangle medians, born on 2008-02-06, modified 2008-02-08.
Object id is 10241, canonical name is CommonPointOfTriangleMedians.
Accessed 2045 times total.

Classification:

AMS MSC:

51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)

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