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.