SOLUTION: Given: triangle ABC, with midpoints at D,E, and F Prove: triangle ADF = triangle FEC

Given: triangle ABC.
       D is midpoint of AB;  E is midpoint of BC;  F is midpoint of AC.

                  C
                  o               Since D is midpoint of AB,  AD = 0.5*AB.          
                 / \        .     Since FE is midline of the triangle ABC, FE = 0.5*AB.
                /   \             Therefore, AD = FE.
               /     \
            F o-------o E         Since E is midpoint of BC, EC = 0.5*BC.
             / \     / \          Since DF is midline of the triangle ABC, DF = 0.5*BC.
            /   \   /   \         Therefore, EC = DF.
           /     \ /     \
          o-------o-------o       Since F is midpoint of AC, AF = FC.
         A        D        B      
                                 

Thus, there are three pairs of congruent sides in triangles ADF and FEC: 

      AD = FE;  DF = EC;  AF = FC.


From it, we conclude that triangles ADF and FEC are congruent, 
due to SSS-test of triangles congruency.


QED.