In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

ABCDis parallelogramUsing properties of a parallelogramAB∥DCandAB=DCEis the mid-point ofABAE=12AB... (1)Fis the mid-point ofCDTherefore,CF=12CDCF=12AB(SinceCD=AB) ... (2)From (1) and (2), we getAE=CFAlso,AE∥CF(SinceAB∥DC)Thus, a pair of opposite sides of a quadrilateralAECFare parallel and equal.QuadrilateralAECFis a parallelogram.EC∥AFEQ∥APandQC∥PFIn△BPA,Eis the mid-point ofBAEQ∥AP∣Using mid point theoremBQ=PQ... (3)Similarly, by taking△CQD, we can prove thatDP=QP... (4)From (3) and (4), we getBQ=QP=PDTherefore,AFandCEtrisect the diagonalBD.