→a,→b,→carecoplanar.⇔[→a→b→c]=0⇔2[→a→b→c]=0......(1)LHS=[→a+→b→b+→c→c+→a]=(→a+→b)⋅[(→b+→c)×(→c+→a)]=(→a+→b)⋅[→b×→c+→b×→a+→c×→c+→c×→a]=(→a+→b)⋅[(→b×→c)−(→a×→b)+(→c×→a)]{as→c×→c=0and→b×→a=−→a×→b}=→a⋅(→b×→c)−→a⋅(→a×→b)+→a⋅(→c×→a)+→b⋅(→b×→c)−→b⋅(→a×→b)+→b⋅(→c×→a)=[→a→b→c]−[→a→a→b]+[→a→c→a]+[→b→b→c]−[→b→a→b]+[→b→c→a]=[→a→b→c]+[→b→c→a]=[→a→b→c]+[→a→b→c]{as[→b→c→a]=[→a→b→c]}=2[→a→b→c]=RHSHence,[→a+→b→b+→c→c+→a]=2[→a→b→c]so,from(1),weget[→a+→b→b+→c→c+→a]=0⇒→a+→b,→b+→c,→c+→aarecoplanar