Triangle Centroid and Position Vectors

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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.