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