InterviewSolution
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InterviewSolution
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In a `/_ABC `points D,E,F are taken on the sides BC,CA and AB respectively such that `(BD)/(DC)=(CE)/(EA)=(AF)/(FB)=n` prove that `/_DEF= (n^2-n+1)/((n+1)^2)` /_ABC` |
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Answer» Take A as the origin and last the postion vectors of point B and C be `vecb and vecc`, Respectively. Therefore, the position vectors, of D, E and F are, respectively, `(nvecc+vecb)/(n+1),vecc/(n+1)and (n vecb)/(n +1)`. Therefore, `vec(ED)=vec(AD)-vec(AE)=((n-1)vecc+vecb)/(n+1)and vec(EF)=(nvecb-vecc)/(n+1)` vector area of `DeltaABC=1/2(vecbxxvecc)` vector area of `DeltaDEF=1/2(vec(EF)xxvec(ED))` `1/(2(n+1)^(2))[(nvecb-vecc)xx{(n-1)vecc+vecb}]` `= 1/(2(n+1)^(2))[(n^(2)-n)vecbxxvecc+vecbxxvecc]` `1/(2(n+1)^(2))[(n^(2)-n+1)(vecbxxvecc)]=(n^(2)-n+1)/((n+1)^(2))Delta_(ABC)` |
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