Points, Lines and Circles: Relative Positions

Try this

Drag $O_{2}$ slowly from the far right to the far left. Which positions does it pass through? Say their names in order.
Watch the number line below. When the blue pointer $d$ reaches the marks $R + r$ and $R - r$ , what is happening in the picture above?
How does the number of “Common points” change along the way? And the number of “Common tangents”?
Change the small radius $r$ . Which marks on the number line move? Does the “intersect” stretch get wider or narrower?

Two circles can sit in five different positions, and which one you get depends only on where the distance between the centers $d$ falls on the number line:

Position	apart	externally tangent	intersecting	internally tangent	one inside the other
$d$ vs. the radii	$d > R + r$	$d = R + r$	$R - r < d < R + r$	$d = R - r$	$d < R - r$
Common points	$0$	$1$	$2$	$1$	$0$

Predict first

Two circles have radii $5$ and $3$ , and their centers are $2$ apart. What is their position?

Externally tangentIntersectingInternally tangentOne inside the other

Choose an answer first; then you can see the result.

Why the boundaries are R + r and R − r

Five positions, one number line

Step 1 / 5

Apart
The circles are far apart. Along the line of centers there is still a gap between them, so $d$ is longer than the two radii put together.

$d > R + r$
Externally tangent
Intersecting
Internally tangent
One inside the other

Points and circles, lines and circles

The idea is exactly the same, except that the number line now has only one boundary point: the radius $r$ .

A point and a circle: let $d$ be the distance from the point to the center. $d < r$ : inside the circle; $d = r$ : on the circle; $d > r$ : outside the circle.
A line and a circle: let $d$ be the distance from the center to the line. $d < r$ : they intersect ( $2$ common points); $d = r$ : tangent ( $1$ ); $d > r$ : they do not meet ( $0$ ).

Drag the line left and right. In how many positions does it have exactly one common point with the circle?

A line with exactly one common point with a circle is called a tangent to the circle. The next stop is all about tangents.

Practice

Exercise 1Basic

A circle has radius $4$ , and the distance from its center to line $l$ is also $4$ . How many common points do line $l$ and the circle have?

Exercise 2Basic

Two circles have radii $7$ and $2$ and are tangent to each other. What can the distance between their centers be? (List every possibility, separated by commas.)

Exercise 3Advanced

Two circles have radii $3$ and $5$ , the distance $d$ between their centers is an integer, and the circles intersect. How many possible values of $d$ are there?

Exercise 4Challenge

Three circles have radii $1$ , $2$ and $3$ , and each pair is externally tangent. Find the area of the triangle whose vertices are the three centers.

Try this

Why the boundaries are R + r and R − r

Five positions, one number line

Points and circles, lines and circles

Practice

Mistake Museum · Exhibits from this stop

Hearing “tangent” and thinking only of external tangency

No common point, so they must be apart