SOLUTION: I need to solve a proof for triangle ABC and DEC trying to prove C is the midpoint of AD. I am given that BC is equal to EC and angle A is equal to angle D

I'm sorry, but that is not true.  Here is why:

Draw an isosceles triangle, label the two base angles A and D



Pick a point on the base of that isosceles triangle that is
obviously NOT its midpoint, and label it C.



With a compass, swing an arc with center C that cuts both legs of the
isosceles triangle.  Label those two points where the arc cuts those legs 
B and E 



Draw radii BC and EC




Now in that figure we have triangles ABC and DEC

BC is equal to EC because 
they are radii of the same arc.

Also angle A is equal to angle D because they are base 
angles of an isosceles triangle.

But C is NOT the midpoint of AD!!!

So what you were asked to prove is just not true!!!

Edwin