In parallelogram ABCD:
AO = OC…………….. (i) (Diagonals of a parallelogram bisect each other)
AE = CF…….. (ii) Given
On subtracting (ii) from (i)
AO – AE = OC – CF
EO = OF ….. (iii)
In ΔDOE and ΔBOF
EO = OF (proved)
DO = OB (Diagonals of a parallelogram bisect each other)
∠DOE = ∠BOF (vertically opposite angles are equal in a parallelogram)
By the rule of SAS congruence ΔDOE ≅ ΔBOF
So, DE = BF (Corresponding parts of congruent triangles)
In ΔBOE and ΔDOF
EO = OF (proved)
DO = OB (diagonals of a parallelogram bisect each other)
∠DOF = ∠BOE (vertically opposite angles are equal in a parallelogram)
By the rule of SAS congruence ΔDOE ≅ ΔBOF
∴ DF = BE (Corresponding parts of congruent triangles)
∴ BFDE is a parallelogram, since one pair of opposite sides are equal and parallel.
