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Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

If G is the...

Question

If $G$ is the centroid of a triangle $A B C$ , prove that $¯ G A + ¯ G B + ¯ G C = 0$ .

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Solution

If a triangle has vertices at $\to a, \to b, \to c$ then GS is given by $(\frac{\to a + \to b + \to c}{3})$
$\to G A = (\to a - \frac{\to a + \to b + \to c}{3})$
$= \frac{2 \to a - \to b - \to c}{3}$
$\to G B = \frac{2 \to b - \to a - \to c}{3}, \to G C = \frac{2 \to c - \to a - \to b}{3}$
$\Rightarrow \to G A + \to G B + \to G C = \frac{1}{3} (0) = 0$ .

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