Given
#"For "Delta ABC " we have"#
#AC=BD," " AD=AE" and "AB^2=AC*BC #
#"To prove " /_BAD=/_CEA#
Proof
#"From given condition "AB^2=AC*BC#
#=>AB^2=AC*BC=BD*BC," since "AC=BD#
#=>(AB)/(BC)=(BD)/(AB).............................(1)#
The relation (1) reveals that
#DeltaABC and Delta ABD" are similar"#
# "Their corresponding angles are"#
#/_BAC=/_ADB and color(red)(/_ACB=/_BAD............(1a))#
And the relation of their corresponding sides is
#(AB)/(BC)=(BD)/(AB)=(AD)/(AC)#
From this relation we now consider
#(BD)/(AB)=(AD)/(AC)#
#=>AD*AB=AC*BD#
#=>AE*AB=AC*AC=AC^2,#
#(" since "AD=AE and BD=AC)#
Now rearranging the above relation we can write
#(AE)/(AC)=(AC)/(AB).........................(2)#
This relation (2) reveals that
#DeltaABC and Delta AEC" are similar"#
# "Their corresponding angles are"#
#/_ABC=/_ACE and color(red)( /_ACB =/_CEA .................(2b))#
Comparing Relation (1a) and (2b) we can get
#color(green)(/_BAD=/_CEA#
Proved