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If `overset(to)(A) , overset(to)(B) " and " overset(to)( c) ` are vectors such that `|overset(to)(B) |=|overset(to)( C ) |` . Prove that `| (overset(to)(A) + overset(to)(B)) xx (overset(to)(A) + overset(to)(C )) | xx (overset(to)(B) xx overset(to)(C )) . (overset(to)(B) + overset(to)( C )) = overset(to)(0)`

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Now `(vec(A) + vec(B)) xx (vec(A) + vec(C ))`
`= vec(A) xx vec(A) + vec(B) xx vec(A) + vec(A) xx vec(C ) + vec(B) xx vec(C )`
`=vec(B) xx vec(A) + vec(A) xx vec(C ) + vec(B) xx vec( C) [:' vec(A) xx vec(A)=0]`
`:. [(vec(A) +vec(B)) xx (vec(A) + vec(C ))] xx (vec(B) xx vec( C))`
`=[(vec(B) xx vec(A) +vec(A) xx vec(C ) + vec(B) xx vec(C )] xx (vec(B) xx vec(C )`
`=(vec(B) xxvec(A)) xx (vec(B) xx vec(C )) +(vec(A) xx vec(C )) xx (vec(B) xx vec(C ))`
`={(vec(B) xx vec(A)) ". " vec(C )} vec(B) - {(vec(B)+vec(A))"." vec(B)} vec(C)`
`+{(vec(A) xx vec(C)) "." vec(C )} vec(B) -{(vec(A) xx vec(C)) "." vec(B) } vec(C)`
` [ :' (vec(a) xx vec(b) ) xx vec(c ) = (vec(a) ". " vec(c )) vec(b) - (vec(b) " ." vec(c )) vec(a)]`
`=[vec(B) vec(A) vec(C )] vec(B) - [vec(A) vec(C) vec(B) ] vec(C)`
`[ :' [vec(a) vec(b) vec(c )] =0` if any two of `vec(a) , vec(b) ,vec(c )` are equal ]
`= [vec(A) vec(C ) vec(B) ] (vec(B) -vec(C))`
Now `[(vec(A) +vec(B)) xx (vec(A) + vec(C ))] xx (vec(B) xx vec( C)) " ." (vec(B) xx vec(C ))`
`=([vec(A) vec(C) vec(B)] {vec(B) -vec(C )}) ". " (vec(B) +vec(C))`
`=[vec(A)vec(C) vec(B)]{(vec(B) -vec(C )) .(vec(B) +vec(C ))}`
`=[(vec(A) vec(C )vec(B)] {|vec(B)|^(2) -|vec(C)|^(2)}=vec(0) [ :' |vec(B)|=|vec(C)|`given ]
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