In triangles ABE and CDE,
AE/EC = BE/ED (Because AE:EC=BE:ED) So triangles ABC and APQ are equiangular.
Then angles BAE = DCE, So AB//DC (Becasue angles BAE = DCE are alternate angles and they are equal)
Also in triangles ADE and BCE,
AE/EC = BE/ED (Because AE:EC=BE:ED) So triangles ADE and BCE are equiangular.
Then angles CBE = ADE, So AD//BC (Becasue angles CBE = ADE are alternate angles and they are equal)
So ABCD is a parallelogram (since pairs of opposite sides are parallel)