As shown in the figure, in the quadrilateral ABCD, points E and F are two points on the diagonal, and be = DF. ① if the quadrilateral aecf is a parallelogram, is the quadrilateral ABCD a parallelogram? Why? ② if the quadrilateral aecf is a diamond, is the quadrilateral ABCD a diamond? Why?

Let AC and BD intersect at point o;
(1) ∵ quadrilateral aecf is a parallelogram,
∴OE=OF,OA=OC,
∵BE=FD,
∴OB=OD．
A quadrilateral ABCD is a parallelogram
(2) ∵ quadrilateral aecf is rhombic,
∴OE=OF,OA=OC,AC⊥BD．
∵BE=FD,
∴OB=OD．
The quadrilateral ABCD is a diamond

As shown in the figure, the known quadrilateral ABCD is a parallelogram, extending Ba to e, extending DC to F, so that be = DF, AF intersects BC to h, CE intersects ad to g, aecf is a parallelogram
Is it a polygon

∵ parallelogram ABCD
Ψ AB parallel CD AB = CD
BA is extended to point E and DC to point F
х AE parallel CF
And ∵ AB = CD be = DF
∴AE=CF
The aecf is a parallelogram

As shown in the figure, it is known that in ▱ ABCD, e and F are two points on the diagonal BD, be = DF, points g and H are on the extension line of Ba and DC respectively, and Ag = ch connects Ge, eh, HF and FG

It is proved that the ∵ quadrilateral ABCD is a parallelogram, ∵ AB = CD, ab ∥ CD, ∵ GBE = HDF, ∵ Ag = ch, ∵ BG = DH, ∵ be = DF, ≌ GBE ≌ HDF. ∥ Ge = HF, ∥ geb = HFD. ∥ GEF = HFE. ≌ Ge ∥ HF. ∥ quadrilateral gehf is a parallelogram

As shown in the figure, it is known that in ▱ ABCD, e and F are two points on the diagonal BD, be = DF, points g and H are on the extension line of Ba and DC respectively, and Ag = ch connects Ge, eh, HF and FG

As shown in the figure, in the quadrilateral ABCD, points E and F are two points on the diagonal, and be = DF. ① if the quadrilateral aecf is a parallelogram, is the quadrilateral ABCD a parallelogram? Why? ② if the quadrilateral aecf is a diamond, is the quadrilateral ABCD a diamond? Why?