In `triangleABC`,E and F are mid-points od sides AB and AC respectively. If BF and CE intersect each other at point O, prove that the `DeltaOBC` and quadrilateral AEOF are equal in area.

Answer

To prove that the area of triangle OBC is equal to the area of quadrilateral AEOF in triangle ABC, where E and F are the midpoints of sides AB and AC respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Midpoints**:
Let E be the midpoint of side AB and F be the midpoint of side AC in triangle ABC.

2. **Draw Line Segments**:
Draw line segments BF and CE. These segments intersect at point O.

3. **Apply the Midpoint Theorem**:
According to the midpoint theorem, since E and F are midpoints, the line segment EF is parallel to side BC and EF = 1/2 BC.

4. **Analyze the Areas of Triangles**:
Since EF is parallel to BC, triangles BCF and BCE share the same base BC and are between the same parallel lines (BC and EF). Therefore, the areas of these triangles are equal:
\[ \text{Area}(BCF) = \text{Area}(BCE) \]

5. **Subtract the Common Area**:
The area of triangle OBC is common in both triangles BCF and BCE. Thus, we can write:
\[ \text{Area}(BCF) - \text{Area}(OBC) = \text{Area}(BCE) - \text{Area}(OBC) \]
This simplifies to:
\[ \text{Area}(BCE) - \text{Area}(OBC) = \text{Area}(BCF) - \text{Area}(OBC) \]
Let’s denote:
\[ \text{Area}(BCF) = A_1, \quad \text{Area}(BCE) = A_2, \quad \text{Area}(OBC) = A_3 \]
Thus, we have:
\[ A_1 - A_3 = A_2 - A_3 \]
This implies:
\[ A_1 = A_2 \]

6. **Use the Median Property**:
Since BF is a median of triangle ABC, it divides triangle ABC into two triangles of equal area:
\[ \text{Area}(BAF) = \text{Area}(BCF) \]

7. **Subtract Areas**:
Now, we can subtract the area of triangle BOE from both sides:
\[ \text{Area}(BCF) - \text{Area}(BOE) = \text{Area}(BAF) - \text{Area}(BOE) \]
The left side gives us the area of triangle OBC:
\[ \text{Area}(OBC) = \text{Area}(BAF) - \text{Area}(BOE) \]

8. **Relate to Quadrilateral AEOF**:
The area of quadrilateral AEOF can be expressed as:
\[ \text{Area}(AEOF) = \text{Area}(BAF) - \text{Area}(BOE) \]
Thus, we have:
\[ \text{Area}(OBC) = \text{Area}(AEOF) \]

### Conclusion:
Therefore, we have proved that the area of triangle OBC is equal to the area of quadrilateral AEOF:
\[ \text{Area}(OBC) = \text{Area}(AEOF) \]