Given the following diagram prove that BD barDC bar BE bar

Given the following diagram, prove that BD bar/DC bar = - BE bar/EC bar

Solution

This is a simple problem of application of Manaleus Theorum and Ceva\'s theorum to a given triangle

From Manaleus Theorum ,

(BE/CE)*(TC/AT)*(AS/SB)=-1

=>BE/CE=-{(AT)*(SB)}/{(AS)*(CT)}   ....1

From Ceva\'s theorum ,

(BD/DC)*(TC/AT)*(AS/SB)=1

=>BD/DC=(SB)*(AT)/(AS)*(TC) -2

From 1 and 2 we get

=>BE/CE=-BD/DC

or , BD/DC=-BE/CE