Line joining the centres of two intersecting circles always bisect their common chord. (True/False).

Answer

To determine whether the statement "The line joining the centers of two intersecting circles always bisects their common chord" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circles and Their Centers**:
Let the two intersecting circles be \( C_1 \) and \( C_2 \) with centers \( O_1 \) and \( O_2 \) respectively. Let the points of intersection of the circles be \( P \) and \( M \). The line joining the centers \( O_1 \) and \( O_2 \) is what we need to analyze.

**Hint**: Draw a diagram to visualize the two circles and their intersection points.

2. **Define the Common Chord**:
The line segment \( PM \) is the common chord of the two circles. We need to prove that the line segment \( O_1O_2 \) bisects \( PM \).

**Hint**: Label the midpoint of \( PM \) as \( O \).

3. **Consider the Triangles**:
We will analyze triangles \( O_1PM \) and \( O_2PM \). Since \( P \) and \( M \) are points on both circles, we can use properties of triangles.

**Hint**: Remember that the lengths \( O_1P \) and \( O_1M \) are radii of circle \( C_1 \), and \( O_2P \) and \( O_2M \) are radii of circle \( C_2 \).

4. **Use the Congruence of Triangles**:
By the Side-Side-Side (SSS) congruence criterion:
- \( O_1P = O_1M \) (radii of circle \( C_1 \))
- \( O_2P = O_2M \) (radii of circle \( C_2 \))
- \( O_1O_2 \) is common to both triangles.

Therefore, triangles \( O_1PM \) and \( O_2PM \) are congruent.

**Hint**: Write down the congruence statement clearly.

5. **Conclude About the Angles**:
Since the triangles are congruent, the angles \( \angle O_1PM \) and \( \angle O_2PM \) are equal. This means that the line \( O_1O_2 \) bisects the angle formed by the lines \( O_1P \) and \( O_2P \).

**Hint**: Use the property of angles in congruent triangles to show that \( PO = OM \).

6. **Prove the Bisection of the Chord**:
Since the angles are equal and the segments are equal, it follows that \( PO = OM \). Thus, the line joining the centers \( O_1O_2 \) bisects the common chord \( PM \).

**Hint**: Summarize your findings to conclude that the statement is true.

### Conclusion:
The statement "The line joining the centers of two intersecting circles always bisects their common chord" is **True**.