ABCD is a parallelogram. FD=DC and DE=2DO. DO=x .Prove that BCEF is a parallelogram ?

Given that ABCD is a parallelogram, we know that opposite sides are equal in length and parallel.

Since FD=DC, we can say that FD=DC=x (since DO=x).

Since DE=2DO, we can say that DE=2x.

Now, let's consider the quadrilateral BCEF.

Since FD=DC=x and DE=2x, we can say that FD=DE.

This means that BC is parallel to EF (since opposite sides are equal in length).

Similarly, since FD=DC=x and DE=2x, we can say that DC=2DE.

This means that BF is parallel to CE (since opposite sides are equal in length).

Therefore, we have shown that BC is parallel to EF and BF is parallel to CE in quadrilateral BCEF.

Since opposite sides are parallel, we can conclude that BCEF is a parallelogram.

Given:

- ABCD is a parallelogram
- FD = DC
- DE = 2DO
- DO = x

To Prove:

- BCEF is a parallelogram

Proof:

Step 1: Prove that BE = CF

- In parallelogram ABCD, we have:
- AB = CD (opposite sides)
- AD = BC (opposite sides)
- Since FD = DC and DE = 2DO, we have:
- OF = DC - DO = DC - x = CD - AD
- OE = DE - DO = 2x - x = x

- Therefore, OF = OA and OE = OB, which implies that BE = CF.

Step 2: Prove that BF = EC

- In parallelogram ABCD, we have:
- AB = CD (opposite sides)
- AD = BC (opposite sides)
- Since FD = DC and DO = x, we have:
- FO = DC - DO = DC - x = AD
- OD = DO = x
- Therefore, FO = AD and OD = x, which implies that BF = EC.

Step 3: Conclude that BCEF is a parallelogram

- Since BE = CF and BF = EC, BCEF is a parallelogram.

Therefore, it is proven that BCEF is a parallelogram.