Assignment 7:

Tangent Circles

by

Margo Gonterman

A line is said to be **tangent** to a given circle if the line only touches the circle once.

Alternatively, a line is said to be tangent to a given circle if it lies at a right angle with the radius of the circle.

A line is called a **secant** line if it meets a given circle twice.

A circle can be tangent to another circle and be either completely inside that circle, or completely outside of it.

When Do Two Circles Intersect?

There are three cases.

Note that D is the distance between the centers of the circles.

is the radius of the smaller circle.

is the radius of the larger circle.

Case 1

The two circles do not intersect.

One circle lies completely outside the other.

In this case,

Case 2

The two circles do not intersect.

The small circle lies completely inside the larger circle.

In this case,

Case 3

The two circles intersect.

In this case,

Construct a Circle Tangent to a Line and a Circle

Start with a circle with center C and a line not through the circle which contains the desired point of tangency, P.

Construct a line perpendicular to the given line at point P.

Also, note that the circle with center C has a radius of r.

Construct a circle with center P and radius r.

Mark the intersection of the circle and perpendicular line and label it Q.

Note, marking the intersection on the same side of the line as the original circle will result in the original circle being completely inside the tangent circle. Marking the intersection on the opposite side of the line from the original circle will result in a tangent circle that does not encompass the original circle.

Construct line segment CQ.

Construct the midpoint of CQ and label it M.

Construct a line perpendicular to CQ through M.

Label the intersection of this line and the perpendicular line at P as point A.

Construct a circle with center A and radius AP.

This circle is tangent to the original circle as well as the line.

Construct a Circle Tangent to Two Circles

Start with two circles with centers A and B such that circle B is contained in circle A.

Construct a point C on circle A.

Construct the ray AC.

Construct a circle with center C and radius equal to the radius of circle B.

Mark the intersection of the ray and the circle as point D.

Note, marking the intersection on the inside of the original circle will result in a different tangent circle than marking the intersection on the outside of the circle. Both of the resulting circles will be tangent to the original circles.

Construct the line BD.

Mark the midpoint of line BD as point M.

Construct a line perpendicular to line BD through point M.

Mark the intersection of the perpendicular line and ray AD as point E.

The circle with center E and radius EC will be tangent to both circle A and circle B at the point C.

Tracing the Center of the Tangent Circle

Note that tracing the center of the tangent circle yields an ellipse with foci A and B.

Recall that an ellipse is a locus of points such that the sum of the distances between the two foci are constant.

To see a GSP animation, click HERE.

Notice when one of the circles is completely outside the other, tracing the center of the tangent circle yields a hyperbola.

Recall that a hyperbola is defined as the locus of points such that the difference between the distance to each point is constant.

To see a GSP animation, click HERE.