Formulas/maths/Circles/Condition for Internal Tangency

Condition for Internal Tangency

Two circles touch internally iff the distance between centres equals the absolute difference of radii. Exactly 1 common tangent.

Derivation

Let circle $C_1$ (centre $O_1$ , radius $r_1$ ) lie inside circle $C_2$ (centre $O_2$ , radius $r_2 > r_1$ ), touching it at exactly one point $T$ .

At $T$ , both radii $O_1T$ and $O_2T$ are perpendicular to the common tangent. Since the circles are on the same side of the tangent, the radii point in the same direction. Therefore $O_1$ lies on segment $O_2T$ :

O_2T - O_1T = O_2O_1 \implies r_2 - r_1 = d

In general (without assuming which is larger): $d = |r_1 - r_2|$ .

Common tangents in this configuration: Exactly 1 tangent — at the point of tangency.

Point of tangency: Divides $O_1O_2$ externally in the ratio $r_1 : r_2$ .

The five cases summarised:

| Condition | Configuration | Common tangents | | --------------- | ------------------------------ | -------------------- | -------------------- | --- | | $d > r_1 + r_2$ | Circles external to each other | 4 | | $d = r_1 + r_2$ | External tangency | 3 | | $| r_1 - r_2 | \lt d \lt r_1 + r_2$ | Circles intersect | 2 | | $d = | r_1 - r_2 |$ | Internal tangency | 1 | | $d \lt | r_1 - r_2 |$ | One inside the other | 0 |

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