Relative Positions of Two Circles
Introduction
When two circles exist in the same plane, they can relate to each other in exactly five different ways. The relationship depends on the distance between their centers compared to their radii.
Let's define our key variables:
- $d$ = distance between the centers of the two circles
- $R$ = radius of the larger circle (or first circle)
- $r$ = radius of the smaller circle (or second circle), where $R \geq r$
The Five Relative Positions
| # | Position | Condition | Common Tangents | Intersection Points |
|---|---|---|---|---|
| 1 | Separated (External) | $d > R + r$ | 4 | 0 |
| 2 | Externally Tangent | $d = R + r$ | 3 | 1 |
| 3 | Intersecting | $R - r < d < R + r$ | 2 | 2 |
| 4 | Internally Tangent | $d = R - r$ | 1 | 1 |
| 5 | One Inside the Other | $d < R - r$ | 0 | 0 |
Special Case: When $d = 0$ and $R = r$, the two circles are concentric (same center). When $d = 0$ and $R = r$, they are actually the same circle!