Two round boundaries have radii 40 and 25 mm. How far apart must their centres be for external tangency? 65 mm, the sum of the radii. But internal tangency also exists, at 15 mm. We must distinguish what is being compared.
A circle is the set of plane points at one fixed positive distance from a centre. A disk includes the circle and its interior. A tangent line shares one point with the circle and is perpendicular to the radius at that point. Prerequisites: distance to a line and triangle inequalities.
Radius, diameter and chord
A segment from centre O to a point on the circle is a radius; the same word denotes its length R. A chord joins two distinct circle points. A chord through the centre is a diameter, of length:
D = 2R.
A diameter is the longest chord. For any other chord AB, O,A,B form a triangle, so AB is less than AO+OB=2R. Not every chord is a diameter, and not every segment from the centre is a radius: the radius must end on the circle. AMSI definitions.
A line: no intersection, tangent or secant
Let h be the distance from O to the line, measured perpendicularly. Compare it with R. If h>R, there are no common points. If h=R, there is one point of tangency T. If h<R, there are two points and the line is a secant.
Why a right angle at tangency? The perpendicular foot is the line’s nearest point to O. If that distance is R, the foot lies on the circle and every other line point is farther than R, hence outside. The line therefore touches precisely there. For a tangent, radius OT is that perpendicular. Euclid III.18.
Two circles: compare sum and difference
Take radii R≥r>0 and centre distance d≥0, all in the same length unit. These cases cover every position:
| Condition | What happens to the circles |
|---|---|
| d>R+r | Externally separated: 0 common points |
| d=R+r | External tangency: 1 point |
| R−r<d<R+r | Intersection: 2 points |
| d=R−r>0 | Internal tangency: 1 point |
| 0≤d<R−r | One inside the other without touching: 0 points |
| d=0 and R=r | Coincident circles: infinitely many points |
Radius, chord and tangent
Externally disjoint
Common points: 0
External tangency
Common points: 1
Two intersections
Common points: 2
Internal tangency
Common points: 1
Nested without tangency
Common points: 0
Concentric, different radii
Common points: 0
The same circle
Common points: ∞
A shared point T off the line of centres forms a triangle with them, of sides R,r,d. Its strict inequalities give R−r<d<R+r; two reflected positions of T give two intersections. At the boundaries the triangle flattens, leaving one tangency point on the line of centres. Internal tangency, external tangency.
Distinct circles cannot share more than two points. With identical centres and radii, however, they are the same circle. This is why the internal-tangency condition includes “>0”. Euclid III.10.
The same centre with unequal radii gives concentric circles without common points. Equal radii with different centres do not make circles identical.
Back to round parts
For radii 40 and 25 mm, external tangency requires d=40+25=65 mm. Internal tangency requires d=40−25=15 mm: for instance, a small disk inside a round opening is displaced until it touches the edge.
At 15<d<65 mm the boundaries intersect at two points. That does not mean filled disks share only two points: their interiors overlap too. Identify the object, circle or disk, before applying the table.