Given: ABCD is a trapezium.
 To Prove: DE/EB = CE /EA
Construction: Draw EF || BA || CD, intersecting AD in F. 

Proof:   FE || AB (Given)   DE/EB = DF/FA [According to basic proportionality theorem] --------- (1)   FE || DC (Given)   CE/EA = DF/FA [According to basic proportionality theorem] --------- (2)   From (1) and (2), we get   DE/EB = CE/EA  So, diagonals of a trapezium divide each other proportionally.

**Proof: Diagonals of a Trapezium Divide Each Other Proportionally**

To prove that the diagonals of a trapezium divide each other proportionally, we will consider the following trapezium:

Let ABCD be a trapezium, where AB || CD. The diagonals of the trapezium intersect at point O.

**Step 1: Establishing Triangles**

We can divide the trapezium into two triangles by drawing a line segment from point A to point C. Let this line segment be AC.

By doing so, we have formed two triangles: triangle AOB and triangle COD.

**Step 2: Observing Parallel Lines**

Since AB || CD (given that it is a trapezium), we can conclude that angle AOB and angle COD are alternate interior angles. Therefore, they are congruent.

**Step 3: Observing Transversals**

As we see in the trapezium, the diagonals AO and CO act as transversals for the parallel lines AB and CD.

**Step 4: Applying Similarity**

By using the property of transversals, we can establish that triangle AOB and triangle COD are similar. This is because angle AOB is congruent to angle COD, and angle OAB is congruent to angle ODC (alternate interior angles).

**Step 5: Proving Proportional Division**

According to the property of similar triangles, corresponding sides of similar triangles are proportional.

In triangle AOB and triangle COD, we can establish the following proportional relationships:

AO/OC = AB/CD
OB/OD = AB/CD

**Step 6: Concluding Proportional Division**

From the above proportional relationships, we can deduce that AO/OC = OB/OD.

Therefore, the diagonals of trapezium ABCD divide each other proportionally at point O.

This completes the proof of the fact that the diagonals of a trapezium divide each other proportionally.