Overview

Tangent circles are a common appearance throughout AMC and AIME geometry. Usually, the most important step is to draw lines from the centers to points of tangency, yielding solvable systems of equations. Some problems can be solved by other strategies, such as right triangles, similar triangles, homothety, and Power of a Point. Note whether tangency is external or internal.

Definitions and Key Ideas

• Two circles are externally tangent if they intersect at exactly one point and neither circle lies inside the other.
• Two circles are internally tangent if they intersect at exactly one point and one circle lies inside the other.
• If two circles are tangent, their centers and point of tangency are collinear, meaning all three points lie on the same line.
• If $d$ is the distance between centers, external tangency gives $d=r_1+r_2$; internal tangency gives $d=|r_1-r_2|$.
• Connect centers and draw radii to points of tangency, and write systems of equations involving right triangles.
• Length of external tangent to two externally tangent circles: $L = 2\sqrt{r_1r_2}.$
• The center of homothety for two externally tangent circles is the tangency point, producing equal length ratios/dilations.

Properties

Distance Between Centers

For externally tangent circles,

For internally tangent circles,

These formulas are often the first step in a solution.

Line of Centers

Suppose circles with centers $O_1,O_2$ are tangent at $P$.

Then

are collinear.

Systems of Equations

It is usually a good idea to draw the radii from centers to points of tangency, and to connect radii of tangent circles, knowing that this segment will also go through the tangent point. Then, frequently you will have to solve a system of equations. Here is an example to illustrate the strategy:

Problem: Circle $O$ has radius $2$ with diameter $AB.$ Circle $C$ has diameter $AO.$ Circle $D$ is internally tangent to circle $O,$ circle $C,$ and diameter $AB.$ Find the radius of circle $D.$

Solution: Notice that circle $C$ must be internally tangent to circle $O,$ and its radius is $\frac{2}{2}=1.$ First, we drop a radius from $D$ to $AB,$ which will be perpendicular to $AB.$ Let the foot be $X.$ Notice that $OD=2-r$ and $DC = 1+r.$ Then,

Expanding and simplifying, we obtain

Therefore, $2OX = 2r+4r-4$ so

Inputting back, we obtain

Therefore, $r=8/9.$

Note: See the section on Descartes' Theorem several modules below; this problem has an alternate solution in which we can duplicate circle $D$ reflected across $AB$ to create the four mutually tangent circle configuration, and then solve for $r$ using the theorem.

Common Tangents and Right Triangles

A very common configuration is two externally tangent circles with a common external tangent.

Let the centers be $O_1, O_2,$ the radii be $r_1,r_2,$ the tangent points be $A$ and $B,$ and let $L$ denote the distance $AB.$ Then drop perpendicular from $O_1$ to $O_2B,$ with foot $X.$ Since $ABXO_1$ is a rectangle (radii to the point of tangency are perpendicular to the tangent line), $O_1X$ will also have length $L.$

This creates right triangle $O_1XO_2,$ so

Therefore,

and so

Homothety

Tangent circles are one of the most common settings in which homothety shows up.

If two circles are externally tangent, the point of tangency is a center of homothety.

Say two circles are externally tangent at point $P.$ Imagine expanding or shrinking one circle about $P,$ that is, that point $P$ stays fixed. Every other point moves directly away from or toward $P$ by the same scale factor, and the points can switch sides of $P$ as well. Eventually, the smaller circle will transform into the larger one. This transformation is called a homothety, and $P$ is called the center of homothety.

Algebraically, suppose a line through $P$ intersects the smaller circle again at $A$ and the larger circle again at $B.$ Then

Because of these equal length ratios, similar triangles will frequently appear because of homothety.

Three Mutually Tangent Circles

Three pairwise tangent circles form one of the most common advanced tangent-circle configurations.

If the radii are $r_1,r_2,r_3,$ then the centers form a triangle with side lengths

Many geometry problems reduce to analyzing this triangle. For instance, using Heron's formula, Law of Cosines, coordinate geometry, etc.

Descartes' Theorem

As shown above, if there is a fourth circle externally or internally tangent to all three, you may also use Descartes' Theorem.

The first thing to calculate is curvature $k_i.$ Curvature is signed, which changes depending on which configuration we have (i.e. whether a circle is internally/externally tangent). Follow the formula

where $r$ is the radius. For four externally tangent circles, all curvatures are positive. When there is one enclosing circle, the three smaller circles have positive curvature, and the big outer circle has negative curvature. Then Descartes' Theorem states:

If $k_4$ is unknown, solve to obtain

Remember to convert curvature back to radius at the end! The larger of the two radii (smaller curvature) generally corresponds to the internally tangent configuration, and the smaller radius to the externally tangent configuration.

Straight Lines

If we replace one of the circles (say $k_3$) with a line, for instance three circles are internally tangent to a line, it can be viewed with a circle of infinite radius, in which case the curvature would be $0.$ Thus, we set $k_3=0$ and so the theorem becomes

We can also square root both sides:

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.

$2+3+3+5=13$.

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 $r=4$.

Common Pitfalls

• Using $r_1+r_2$ for internal tangency.
• Confusing center distance or radius with a diameter.
• Forgetting to calculate curvature or convert curvature to radius at the end of Descartes' Theorem.
• Getting confused with signs/absolute values.

Practice Problems