If two circles intersect at 2 points prove that their centres lie on t...
If two circles intersect at two points, then they share a common chord connecting those two points. The perpendicular bisector of this common chord is a line that passes through the midpoint of the chord and is perpendicular to the chord.
To prove that the centers of the two circles lie on the perpendicular bisector of the common chord, we can use the following steps:
1. Draw the common chord and the centers of the two circles.
2. Draw the midpoint of the common chord.
3. Draw the perpendicular bisector of the common chord, passing through the midpoint and perpendicular to the chord.
4. Observe that the centers of the two circles lie on the perpendicular bisector of the common chord.
This can be proven by the fact that the perpendicular bisector of a chord passes through the center of the circle. Since the centers of the two circles lie on the perpendicular bisector, they must be the center of the circle.
Therefore, if two circles intersect at two points, their centers lie on the perpendicular bisector of the common chord connecting those two points.