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Question

Show that a,b,c are coplanar if and only if (a+b),(b+c) and (c+a) are coplanar.

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Solution

a,b,carecoplanar.[abc]=02[abc]=0......(1)LHS=[a+bb+cc+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)]{asc×c=0andb×a=a×b}=a(b×c)a(a×b)+a(c×a)+b(b×c)b(a×b)+b(c×a)=[abc][aab]+[aca]+[bbc][bab]+[bca]=[abc]+[bca]=[abc]+[abc]{as[bca]=[abc]}=2[abc]=RHSHence,[a+bb+cc+a]=2[abc]so,from(1),weget[a+bb+cc+a]=0a+b,b+c,c+aarecoplanar


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