Question Number 103774 by dw last updated on 17/Jul/20

Answered by mr W last updated on 17/Jul/20

METHOD I AB^(→) =a BC^(→) =b CA^(→) =c a+b+c=0 AH=a+(1/9)b BG=−a−(1/3)c CP=−b−(5/6)a BE=λBG=−λa−(λ/3)c AE=μAH=μa+(μ/9)b=a−λa−(λ/3)c μa+(μ/9)b=(1−λ)a+(λ/3)(a+b) μa+(μ/9)b=(1−(2/3)λ)a+(λ/3)b ⇒μ=3λ ⇒μ=3λ=1−(2/3)λ ⇒λ=(3/(11)) ⇒μ=(9/(11)) AF=ξAH=ξa+(ξ/9)b CF=νCP=−νa−((5ν)/6)b=CA+AF −νb−((5ν)/6)a=−a−b+ξa+(ξ/9)b −((5ν)/6)b−νa=(ξ−1)a+((ξ/9)−1)b ⇒−ν=(ξ/9)−1 ⇒−((5ν)/6)=ξ−1 ⇒(5/6)((ξ/9)−1)=ξ−1 ⇒ξ=(9/(49)) ⇒ν=((48)/(49)) BD=ηBG=−ηa+(η/3)(a+b)=−((2η)/3)a+(η/3)b CD=εCP=−εb−((5ε)/6)a=CB+BD −εb−((5ε)/6)a=−((2η)/3)a+((η/3)−1)b ⇒−((5ε)/6)=−((2η)/3) ⇒5ε=4η ⇒−ε=(η/3)−1 ⇒η=((15)/(17)) ⇒ε=((12)/(17)) ED=BD−BE=(η−λ)BG= =(((15)/(17))−(3/(11)))BG=((114)/(187))BG EG=BG−BE=(1−λ)BG=(8/(11))BG ((ED)/(EG))=((114)/(187))×((11)/8)=((57)/(68)) EF=AE−AF=(μ−ξ)AH =((9/(11))−(9/(49)))AH=((342)/(539))AH ((EF)/(EA))=((342)/(539))×(1/μ)=((342)/(539))×((11)/9)=((38)/(49)) S_(DEF) =((EF)/(EA))×((ED)/(EG))×S_(AEG) =((EF)/(EA))×((ED)/(EG))×((EG)/(BG))×S_(ABG) =((EF)/(EA))×((ED)/(EG))×((EG)/(BG))×((AG)/(AC))×S_(ABC) =((38)/(49))×((57)/(68))×(8/(11))×(1/3)×S_(ABC) =((1444)/(9163))×S_(ABC) ⇒(S_(DEF) /S_(ABC) )=((1444)/(9163))≈0.1576