Overview

Tangent circles convert to center-distance equations. Decide whether tangency is external or internal.

Core Skills

Identify Tangency Type

External tangency gives $d=r_1+r_2$; internal tangency gives $d=|r_1-r_2|$.

Use a Coordinate Setup

Place centers on a line when possible to reduce to one-dimensional distances.

Combine with Power of a Point

If a circle is tangent to a line or another circle, use power-of-point to connect lengths.

Key Ideas

• External tangency: distance between centers equals $r_1 + r_2$.
• Internal tangency: distance equals $|r_1 - r_2|$.

Worked Example

Two circles of radii $5$ and $3$ are externally tangent. Find the distance between centers.

Distance is $5+3=8$.

More Examples

Example 1: Internal Tangency

Two circles with radii $10$ and $4$ are internally tangent. Find the distance between centers.

$|10-4|=6$.

Example 2: Tangent Chain

Three circles with radii $2,3,5$ are tangent in a line. Find the distance between the centers of the first and third.

$2+3+5=10$.

Example 3: Mixed Condition

If a circle of radius $r$ is tangent internally to a circle of radius $9$ and the centers are $5$ units apart, find $r$.

$9-r=5$, so $r=4$.

Strategy Checklist

• Decide external vs internal tangency first.
• Translate to center distances.
• Use absolute values for internal tangency.

Common Pitfalls

• Using $r_1+r_2$ for internal tangency.
• Forgetting to take absolute value for internal tangency.
• Confusing center distance with a diameter.

Practice Problems