Question 934601
AC=BD, then AB=CD


{{{drawing(300,280,-15,15,-6,6, line(-10,0,10,0),
   line(-1.5,0,0,0),
   line(1.5,0,10,0),
circle(-10,0,.25),
circle(-1.5,0,.25),
circle(1.5,0,.25),
circle(10,0,.25),
   locate(-10,0,A), locate(-1.5,0,B), locate(1.5,0,C), locate(10,0,D)
 )}}} 


 From the above figure  we get that
{{{AC     = AB + BC}}}
{{{BD     = BC + CD}}}

It is given that {{{AC = BD}}}- if left sides equal, then right sides are equal too; so, we get:

{{{AB + BC      = BC + CD}}}            … (1)

According to Euclid’s axiom,when equals are subtracted from equals, the remainders are also equal.

Subtracting {{{BC}}} from both side in equation (1), we get

{{{AB + BC-BC    = BC + CD -BC}}}

{{{AB     = CD}}} .........=>Hence proved.