Find the centres and radii \(r_1,r_2>0\), then calculate the centre distance \(d\).
\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
The following five cases concern distinct circles. If \(d=0\) and \(r_1=r_2\), the circles coincide.
Separate: \(d>r_1+r_2\). No common point.
Externally tangent: \(d=r_1+r_2\). One common point.
Intersecting: \(|r_1-r_2|<d<r_1+r_2\). Two common points.
Internally tangent: \(d=|r_1-r_2|\). One common point.
One circle inside the other: \(d<|r_1-r_2|\). No common point.
Internal tangency requires \(d=|r_1-r_2|>0\). For two intersections, both strict inequalities must hold.
| Condition on \(d\) | Relative position | Common points |
|---|---|---|
| \(d>r_1+r_2\) | Each circle lies entirely outside the other | 0 |
| \(d=r_1+r_2\) | Externally tangent (touch from outside) | 1 |
| \(|r_1-r_2|<d<r_1+r_2\) | Intersect at two distinct points | 2 |
| \(d=|r_1-r_2|\) | Internally tangent (touch from inside) | 1 |
| \(d<|r_1-r_2|\) | One circle lies inside the other | 0 |
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