Prove that the line through the two points d and e is the perpendicular bisector of the segment

Draw a line through the points D and E to meet AB and F.

Consider triangles ADE and BDE. 
Side DE is common to both triangles. and AE=BE and AD=BD (both given)
Therefore the 2 triangles are congruent by SSS and so m<ADE = m<BDE.

In a similar way triangles AEF and BEF are congruent and so
m <AEF = m<BEF.

So we see that the line DEF bisects both  angles ADB and AEB  and since both triangles ADB and AEB are isosceles then DE is perpendicular to AB.

Ihave, but this question is gonna probably get deleted.