# Thread: need help prove Medians of a triangle coincident

1. ## need help prove Medians of a triangle coincident

show that the medians of a triangle ABC are coincident

2. Originally Posted by Reignrx
How can i "show that the medians of a triangle ABC are coincident (at point X). and "What is the value of A'X/AX"?
This is a completely standard theorem. Its proof is in any standard textbook.
There is a very straightforward proof using vectors.

3. thanks

4. Let $\displaystyle \overrightarrow a,\overrightarrow b,\overrightarrow c$, denote the vertices $\displaystyle A,B,C$ respectively and also assume $\displaystyle D(\overrightarrow d),E(\overrightarrow e),F(\overrightarrow f)$ denote the mid-points of $\displaystyle AB,BC,CA$ respectively.Therefore,

$\displaystyle \overrightarrow d=\frac{\overrightarrow b+\overrightarrow c}{2}$

$\displaystyle \overrightarrow e=\frac{\overrightarrow c+\overrightarrow a}{2}$

$\displaystyle \overrightarrow f=\frac{\overrightarrow a+\overrightarrow b}{2}$

$\displaystyle 2\overrightarrow d=\overrightarrow b+\overrightarrow c; 2\overrightarrow e=\overrightarrow c+\overrightarrow a; 2\overrightarrow f=\overrightarrow a+\overrightarrow b$

$\displaystyle 2\overrightarrow d+\overrightarrow a=2\overrightarrow e+\overrightarrow b=2\overrightarrow f+\overrightarrow c=\overrightarrow a+\overrightarrow b+\overrightarrow c$

$\displaystyle \frac{2\overrightarrow d+\overrightarrow a}{3}= \frac{2\overrightarrow e+\overrightarrow b}{3}= \frac{2\overrightarrow f+\overrightarrow c}{3}=\frac{\overrightarrow a+\overrightarrow b+\overrightarrow c}{3}$

Now,$\displaystyle \frac{2\overrightarrow d+\overrightarrow a}{3}$ is the position vector of the point on median $\displaystyle AD$ which divides $\displaystyle AD$ in the ratio $\displaystyle 2:1$.Simlarly,$\displaystyle \frac{2\overrightarrow e+\overrightarrow b}{3}$ is the position vector of the point on median $\displaystyle BE$ which divides $\displaystyle BE$ in the ratio $\displaystyle 2:1$ and $\displaystyle \frac{2\overrightarrow f+\overrightarrow c}{3}$ is the position vector of the point on median $\displaystyle CF$ which divides $\displaystyle CF$ in the ratio $\displaystyle 2:1$.

Since these position vectors have turned out to be equal,obviously $\displaystyle AD,BE\\,and\\,CF$ are concurrent and the point of concurrency is $\displaystyle \frac{\overrightarrow a+\overrightarrow b+\overrightarrow c}{3}$

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# prove by vector method that the medians of a triangle are coincident.

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