Proof:

1. Introduction
Let's consider two circles, C1 and C2, that intersect each other at two points. We need to prove that the line passing through their centers is perpendicular to the common chord.

2. Definitions
Before we proceed with the proof, let's define some important terms:
- Circle: A circle is a closed curve formed by all the points in a plane that are equidistant from a fixed center point.
- Center: The center of a circle is the point from which all points on the circle are equidistant.
- Chord: A chord is a line segment that connects two points on a circle.
- Common Chord: The common chord is the line segment that is common to both circles C1 and C2.

3. Construction
To prove that the line passing through the centers of the two circles is perpendicular to the common chord, we need to construct the figure as follows:
- Draw two circles, C1 and C2, intersecting each other at points A and B.
- Let O1 and O2 be the centers of circles C1 and C2, respectively.
- Draw the common chord AB.

4. Proof
To prove that the line passing through the centers of the two circles is perpendicular to the common chord, we will use the concept of perpendicular bisectors.

- Let M be the midpoint of the common chord AB.
- Draw a line segment OM, connecting the centers O1 and O2.
- Since O1 and O2 are the centers of circles C1 and C2, respectively, OM will pass through the midpoints of chords AB in both circles.
- By definition, the perpendicular bisector of a chord passes through the center of the circle.
- Therefore, OM is the perpendicular bisector of chord AB in both circles.

5. Conclusion
From the above proof, we can conclude that the line passing through the centers of the two circles is perpendicular to the common chord.