As shown in the figure, ab | DC, ad | BC, points E and F are on AB and DC respectively, and be = DF, AF = CE

As shown in the figure, ab | DC, ad | BC, points E and F are on AB and DC respectively, and be = DF, AF = CE

AF = CE for the following reasons
∵AB||DC,AD||BC
The quadrilateral ABCD is a parallelogram
∴CD=AB,∠D=∠B,AD=CB
In △ ADF and △ CBE
Ad = CB (proved)
{d = ∠ B (proved)
{DF = be (known)
∴△ADF≌△CBE（SAS）
‖ AF = Ce (the corresponding sides of congruent triangles are equal)

As shown in the figure, ∠ 1 = ∠ 2, DC = BC, CE ⊥ AB, CF ⊥ ad, verify, be = DF

Because in RT △ AFC and RT △ AEC,
{angle 1 = angle 2
{AC=AC
So RT △ AFC ≌ RT △ AEC
So CF = CE
Because in RT △ DFC and RT △ BEC,
{DC=BC
{CF=CE
So RT △ DFC ≌ RT △ bec
So be = CF