In an isosceles `triangleABC`, where, AC=BC. A line segment DE is drawn parallel to the side AB, where D and E are midpoints od AC and BC, respectively. The disgonals AE and DB of the trapezium ABED intersect each other at O. which of the following are necessarily true?
`Area triangleDOA=Area triangleEOB`
`Area triangleAOB=4(Area triangleDOE)`
`Area triangleCDE=9(Area triangleDOE)`

Options

A. A. (i), (ii) and (iii)
B. B. (ii) and (iii)
C. C. (i) and (iii)
D. D. (I) and (ii)

Answer

To solve the problem, we need to analyze the properties of the isosceles triangle ABC and the trapezium ABED formed by the midpoints D and E of sides AC and BC, respectively. We will also look at the areas of the triangles formed by the intersection of the diagonals AE and DB at point O.

### Step 1: Understand the Configuration
We have an isosceles triangle ABC where AC = BC. D and E are midpoints of AC and BC, respectively, and DE is drawn parallel to AB. The diagonals AE and DB intersect at point O.

### Step 2: Prove Area of Triangle DOA = Area of Triangle EOB
Since DE is parallel to AB, triangles DOA and EOB are similar by the AA (Angle-Angle) criterion. The angles at O are equal because they are alternate interior angles formed by the transversal AB cutting the parallel lines DE and AB.

- Since D and E are midpoints, the lengths of segments AD and BE are equal.
- Therefore, by the properties of similar triangles, the area of triangle DOA is equal to the area of triangle EOB.

**Conclusion:** Area of triangle DOA = Area of triangle EOB (True).

### Step 3: Prove Area of Triangle AOB = 4 * Area of Triangle DOE
Triangles AOB and DOE are also similar. The ratio of their corresponding sides is 2:1 because DE is half the length of AB (since D and E are midpoints).

- The area of similar triangles is proportional to the square of the ratio of their sides. Therefore, if the ratio of the sides is 2:1, the ratio of the areas will be (2^2):(1^2) = 4:1.
- Thus, Area of triangle AOB = 4 * Area of triangle DOE.

**Conclusion:** Area of triangle AOB = 4 * Area of triangle DOE (True).

### Step 4: Prove Area of Triangle CDE = 9 * Area of Triangle DOE
Now we need to analyze triangles CDE and DOE. Triangle CDE is not similar to triangle DOE, as they do not share the same angles.

- Triangle CDE is part of triangle ABC, and since D and E are midpoints, triangle CDE is similar to triangle CAB, but not directly proportional to triangle DOE.
- The area of triangle CDE cannot be 9 times the area of triangle DOE because the ratio of the areas of triangle CDE to triangle DOE is not established as 9:1 based on the previous analysis.

**Conclusion:** Area of triangle CDE = 9 * Area of triangle DOE (False).

### Final Results
- Statement 1: True (Area triangle DOA = Area triangle EOB)
- Statement 2: True (Area triangle AOB = 4 * Area triangle DOE)
- Statement 3: False (Area triangle CDE ≠ 9 * Area triangle DOE)