Question

If g is the centroid of triangle abc prove that ga+gb+gc=0

Answer 1

Here is the answer to your question:

If G(g↑) is the centroid of a triangle ABC, then


... (g↑) = ( a↑ + b↑ + c↑ ) / 3,

i.e., ( a↑ + b↑ + c↑ ) = 3(g↑) ............ (1)
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Then,

... (GA↑) + (GB↑) + (GC↑)

= ( a↑ - g↑ ) + ( b↑ - g↑ ) + ( c↑ - g↑ )

= ( a↑ + b↑ + c↑ ) - 3(g↑)

= 3(g↑) - 3(g↑) .................... from (1)

= (0↑)

Hence, the result.

Answer 2

If G is the centroid of a triangle ABC, then

... G = ( a + b + c ) / 3,

i.e., ( a + b + c ) = 3G ............ (1)

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Then,

... (GA) + (GB) + (GC)

= ( a - G ) + ( b - G ) + ( c - G )

= ( a + b+ c ) - 3G

= 0

Hence, the result.

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