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

Find the centres and radii \(r_1,r_2>0\), then calculate the centre distance \(d\).

The following five cases concern distinct circles. If \(d=0\) and \(r_1=r_2\), the circles coincide.

Internal tangency requires \(d=|r_1-r_2|>0\). For two intersections, both strict inequalities must hold.

Each circle lies entirely outside the other

Relative Positions of Two Circles

Relative Positions of Two Circles