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We can treat points in a plane as

**position vectors**(or simply vectors). If you have not heard about vectors before, they are mathematical objects that we can perform addition and scalar multiplication (i.e. scaling).

For instance, given two points

*A*and

*B*, the point

*X*which lies on the line joining

*A*and

*B*such that

*AX:XB=r:1-r*can be written as

*X=(1-r)A+rB*

Note that

*r*can be any number. If

*r<0*

*then**X*lies on the*BA*extended; if*0* then *X*lies on the line segment

*AB*; if

*r>1*then

*X*lies on the

*AB*extended.

*In particular,*

*0.5A+0.5B*is the mid-point of

*AB*.

Now, given three points

*A*,

*B*and

*C*. Consider

*M=1/3 A+1/3 B+1/3 C*

Now

*M=1/3 A+(2/3)(0.5B+0.5C)*, and so

*M*indeed lies on the median joining

*A*and the mid-point of

*BC*. Likewise, it lies on the other two medians.

Therefore

*M*is the centroid of the triangle

*ABC*. Moreover, from the formula, we can tell that it always divides the medians in the ratio 2:1.