Two ⊙ — Two-circle position relations · Principia

Depends on: Ruler axiom (distance between two points), Triangle inequality a + b > c.

Statement

Let $\odot O_1(r_1)$ and $\odot O_2(r_2)$ be two distinct circles, and write $d=|O_1O_2|$ for the distance between their centers. Marking the two critical values $r_1+r_2$ and $|r_1-r_2|$ on a number line, the positional relationship between the two circles falls into exactly 5 cases according to which interval $d$ lies in:

\underbrace{d>r_1+r_2}_{\text{externally separate}} \quad \underbrace{d=r_1+r_2}_{\text{externally tangent}} \quad \underbrace{|r_1-r_2|<d<r_1+r_2}_{\text{intersecting}} \quad \underbrace{d=|r_1-r_2|}_{\text{internally tangent}} \quad \underbrace{d<|r_1-r_2|}_{\text{one contained in the other}}

The number of common points is $0,1,2,1,0$ respectively; the point of tangency (external or internal) must lie on the line of centers $O_1O_2$ .

$Two-circle position relations: d landing at different points of [\,|r_1-r_2|,\;r_1+r_2\,] corresponds to the 5 configurations$

Two-circle position relations (5 cases)

Statement

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