Cambridge O Level Additional Mathematics · Syllabus 4037 · Coordinate Geometry of the Circle
Line of Centres
What is Line of Centres?
The straight line joining the centres of two circles, and the axis of symmetry of the whole configuration. Its length is the distance d between the centres, and comparing d with the sum and the difference of the radii decides completely how the two circles are positioned: they cut at two points when d lies strictly between the difference and the sum of the radii, they touch at one point when d equals the sum, giving external contact, or equals the non-zero difference, giving internal contact, and they have no common point when d exceeds the sum or falls below the difference. When two circles touch, the single point of contact always lies on this line, which is what makes the contact point straightforward to compute. When two circles cut at two points, the line of centres is the perpendicular bisector of their common chord.
This definition is part of the Coordinate Geometry of the Circle chapter in Cambridge O Level Additional Mathematics.