Question

cho m, n , p lần lượt nằm trên cạnh bc , ca , ab của tam giác abc biết am, bn , cp đồng quy tại e . cmr :ae/em = an/cn + ap/bp

Answer 1

\(\dfrac{AE}{EM}=\dfrac{S_{AEC}}{S_{MEC}}=\dfrac{S_{AEB}}{S_{MEB}}=\dfrac{S_{AEC}+S_{AEB}}{S_{BEC}}\)

\(\dfrac{AN}{BN}=\dfrac{S_{AEN}}{S_{CEN}}=\dfrac{S_{ABN}}{S_{CBN}}=\dfrac{S_{ABN}-S_{AEN}}{S_{CBN}-S_{CEN}}=\dfrac{S_{AEB}}{S_{BEC}}\)

\(\dfrac{AP}{BP}=\dfrac{S_{AEP}}{S_{BEP}}=\dfrac{S_{ACP}}{S_{BCP}}=\dfrac{S_{ACP}-S_{AEP}}{S_{BCP}-S_{BEP}}=\dfrac{S_{ACE}}{S_{BEC}}\)

\(\Rightarrow\dfrac{AN}{BN}+\dfrac{AP}{BP}=\dfrac{S_{AEB}+S_{ACE}}{S_{BEC}}=\dfrac{AE}{EM}\)