TANGENT CIRCLES

by Laurel A. Bleich and Eugenia Vomvoridi

Task: Given two circles and a point on one of the circles, construct a circle tangent to the
two circles with one point of tangency being the designated point.

Problem: What shape of the locus would you expect to find when you trace the center of the tangent circle?

WHEN ONE CIRCLE IS LOCATED INSIDE THE OTHER

CASE 1: When the tangent circle is located inside the larger circle and does not contain the small circle.

First construct the given tanget Circle:

Given: The center of the desired circle will lie along a line from the center of the given circles with the specified point.

Take the radius of the smaller circle (AB) and construct another circle with the same radius measure centered at point D.

Construct a segment from point A (center of the smaller cirlce) and point E - point of intersection of the constructed circle and line through the center of the larger circle. Then construct the perpendicular bisector for this segment.

Why does this work? By construction, point G is the midpoint of segment AE. Therefore |AG| = |GE|. Additionally, GF is perpendicular to AE creating two congruent triangles (triangle AGF and triangle EGF). The |AF| = |EF|. Since by construction, |AB| = |DE|, it follows that |BF| = |DF|. By creating a circle with center F whose radius has length |BF|, we will always have a circle tangent to both circles at the points B and D.

Find the point of Intersection with the perpendicular bisector and the line through the center. This becomes the center of the tangent circle.

Construct a circle centered at point F, with a radius of FD. (Recall point D is the point we want the constructed circle to be tangent to)

Question: What if E was a different point? Would the same arguement hold?

What shape is the locus of the newly created tangent circle?

CLICK HERE to view the animated trace of the center of the tangent circle.

The locus appears to be an ellipse.

Proof that the locus is an ellipse with foci points A and C which are the centers of the two original circles:

|AF| = |FE| by congruency of triangles.

|FD| = |FC| which is the radius of the big circle.

|AB| = |DE| which is the radius of the small circle.

Therefore, |AF| + |FC| = |FD| + |FE| = |FC| + |FE| = |CD| +|DE|, which is constant because it is the radius of the big circle plus the radius of the small circle.

By the geometric definition of the ellipse, the locus created is an ellipse.

CASE 2: When the tangent circle is located inside the larger circle and does contain the small circle.

Similarly, we construct a tangent circle in this case with the diffence that the point E is inside the larger circle rather than outside.

Click Here for an animation.

Notice that the locus is an ellipse as well.

|CF| + |AF| = |CF| + |FE| = |CD| - |DE| which is constant because it is the difference of the radii of the original circles.

PROBLEM: Explore the case when the two original circles are concentric (that is they have the same center). What would you expect the locus to be?

WHEN THE SMALLER CIRCLE IS LOCATED OUTSIDE THE LARGER CIRCLE

GIVEN TWO CIRCLES DISJOINT, CONSTRUCT A TANGENT CIRCLE TO BOTH

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

What is the locus of the center of the tangent circle?

CLICK HERE to animate the creation of the locus.

The locus seems to be a hyperbola. In let's proof that is it a hyperbola with foci points A and C.

PROOF: |AF| = |EF| by construction.

|DE| = |AB| by construction.

|AF| - |FC| = |EF| - |FC| = |EC| = |ED| + |CD| which is the sum of the radii of the original circles which is a constant.

Therefore, by the geometric definition of a hyperbola, the locus is a hyperbola.

Explore other cases in which the two circles are disjoint. Will the locus always be a hyperbola?

WHEN THE TWO CIRCLES INTERSECT

GIVEN TWO CIRCLE INTERSECTING, CONSTRUCT A TANGENT CIRCLE

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

What is the locus of the center of the tangent circle?

CLICK HERE to animate the creation of the locus.

The locus appears to be an ellipse with foci A and C.

Proof:

|AF| = |FE| by construction

|AB| = |DE| by construction

|AF| + |FC| = |EF| + |FC| = |CE| = |CD| + |DE| = |CD| + |AB|, which is the sum of the sum of the radii of original circles.

By the geometric definition of the ellipse, the locus is an ellipse with foci A and C, the centers of the original circles.

Explore the case when the two circles are tangent. What is the locus?

Return to Eugenia's Home Page
Return to Laurel's Home Page


Given: The center of the desired circle will lie along a line from the center of the given circles with the specified point.	Take the radius of the smaller circle (AB) and construct another circle with the same radius measure centered at point D.	Construct a segment from point A (center of the smaller cirlce) and point E - point of intersection of the constructed circle and line through the center of the larger circle. Then construct the perpendicular bisector for this segment.

		Why does this work? By construction, point G is the midpoint of segment AE. Therefore \|AG\| = \|GE\|. Additionally, GF is perpendicular to AE creating two congruent triangles (triangle AGF and triangle EGF). The \|AF\| = \|EF\|. Since by construction, \|AB\| = \|DE\|, it follows that \|BF\| = \|DF\|. By creating a circle with center F whose radius has length \|BF\|, we will always have a circle tangent to both circles at the points B and D.
Find the point of Intersection with the perpendicular bisector and the line through the center. This becomes the center of the tangent circle.	Construct a circle centered at point F, with a radius of FD. (Recall point D is the point we want the constructed circle to be tangent to)	Question: What if E was a different point? Would the same arguement hold?


GIVEN TWO CIRCLES DISJOINT, CONSTRUCT A TANGENT CIRCLE TO BOTH	THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

TANGENT CIRCLES

by Laurel A. Bleich and Eugenia Vomvoridi

Task: Given two circles and a point on one of the circles, construct a circle tangent to the two circles with one point of tangency being the designated point.

Problem: What shape of the locus would you expect to find when you trace the center of the tangent circle?

WHEN ONE CIRCLE IS LOCATED INSIDE THE OTHER

CASE 1: When the tangent circle is located inside the larger circle and does not contain the small circle.

First construct the given tanget Circle:

Given: The center of the desired circle will lie along a line from the center of the given circles with the specified point.

Take the radius of the smaller circle (AB) and construct another circle with the same radius measure centered at point D.

Construct a segment from point A (center of the smaller cirlce) and point E - point of intersection of the constructed circle and line through the center of the larger circle. Then construct the perpendicular bisector for this segment.

Find the point of Intersection with the perpendicular bisector and the line through the center. This becomes the center of the tangent circle.

Construct a circle centered at point F, with a radius of FD. (Recall point D is the point we want the constructed circle to be tangent to)

Question: What if E was a different point? Would the same arguement hold?

What shape is the locus of the newly created tangent circle?

CLICK HERE to view the animated trace of the center of the tangent circle.

The locus appears to be an ellipse.

Proof that the locus is an ellipse with foci points A and C which are the centers of the two original circles:

|AF| = |FE| by congruency of triangles.

|FD| = |FC| which is the radius of the big circle.

|AB| = |DE| which is the radius of the small circle.

Therefore, |AF| + |FC| = |FD| + |FE| = |FC| + |FE| = |CD| +|DE|, which is constant because it is the radius of the big circle plus the radius of the small circle.

By the geometric definition of the ellipse, the locus created is an ellipse.

CASE 2: When the tangent circle is located inside the larger circle and does contain the small circle.

Similarly, we construct a tangent circle in this case with the diffence that the point E is inside the larger circle rather than outside.

Click Here for an animation.

Notice that the locus is an ellipse as well.

|CF| + |AF| = |CF| + |FE| = |CD| - |DE| which is constant because it is the difference of the radii of the original circles.

PROBLEM: Explore the case when the two original circles are concentric (that is they have the same center). What would you expect the locus to be?

WHEN THE SMALLER CIRCLE IS LOCATED OUTSIDE THE LARGER CIRCLE

GIVEN TWO CIRCLES DISJOINT, CONSTRUCT A TANGENT CIRCLE TO BOTH THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

GIVEN TWO CIRCLES DISJOINT, CONSTRUCT A TANGENT CIRCLE TO BOTH

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

What is the locus of the center of the tangent circle?

CLICK HERE to animate the creation of the locus.

The locus seems to be a hyperbola. In let's proof that is it a hyperbola with foci points A and C.

PROOF: |AF| = |EF| by construction.

|DE| = |AB| by construction.

|AF| - |FC| = |EF| - |FC| = |EC| = |ED| + |CD| which is the sum of the radii of the original circles which is a constant.

Therefore, by the geometric definition of a hyperbola, the locus is a hyperbola.

Explore other cases in which the two circles are disjoint. Will the locus always be a hyperbola?

WHEN THE TWO CIRCLES INTERSECT

GIVEN TWO CIRCLE INTERSECTING, CONSTRUCT A TANGENT CIRCLE THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

GIVEN TWO CIRCLE INTERSECTING, CONSTRUCT A TANGENT CIRCLE

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

What is the locus of the center of the tangent circle?

CLICK HERE to animate the creation of the locus.

The locus appears to be an ellipse with foci A and C.

Proof:

|AF| = |FE| by construction

|AB| = |DE| by construction

|AF| + |FC| = |EF| + |FC| = |CE| = |CD| + |DE| = |CD| + |AB|, which is the sum of the sum of the radii of original circles.

By the geometric definition of the ellipse, the locus is an ellipse with foci A and C, the centers of the original circles.

Explore the case when the two circles are tangent. What is the locus?

Return to Eugenia's Home Page Return to Laurel's Home Page

Task: Given two circles and a point on one of the circles, construct a circle tangent to the
two circles with one point of tangency being the designated point.

GIVEN TWO CIRCLES DISJOINT, CONSTRUCT A TANGENT CIRCLE TO BOTH

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

GIVEN TWO CIRCLE INTERSECTING, CONSTRUCT A TANGENT CIRCLE

THE CONSTRUCTION OF THE TANGENT IS DONE IN A SIMILAR MANNER AS ABOVE.

Return to Eugenia's Home Page
Return to Laurel's Home Page