如图，已知AB∥EF∥CD，若AB=a，CD=b，EF=c，求证：1c=1a+1b．

Question

如图，已知AB∥EF∥CD，若AB=a，CD=b，EF=c，求证：(1)/(c)=(1)/(a)+(1)/(b)．

Answer 1

∵AB∥EF∥CD，
∴△DEF∽△DAB，△BEF∽△BCD，
∴(EF)/(AB)=(DF)/(BD)，(EF)/(CD)=(BF)/(BD)，
∴(EF)/(AB)+(EF)/(CD)=(DF)/(BD)+(BF)/(BD)=1，
∴(1)/(AB)+(1)/(CD)=(1)/(EF)，
∵AB=a，CD=b，EF=c，
∴(1)/(c)=(1)/(a)+(1)/(b)．