Assignment 7 - Tangent Circles

Faith Hoyt

To start off, first let's define what we are wanting to do in this investigation:

We are given two circles and a point on one of these circles. We want to construct a circle that is tangent to both given circles, where our given point is one of the points of tangency. So, how do we go about doing this?

Let's first define what it means to be tangent. As we know, in order for two objects to be tangent to each other, that means that they touch only at one point. However, how do we know how to find this one point? To figure this out we can look at the simple case of one circle and trying to find the line tangent to that circle. (the skeleton of this construction was taken from www.mathopenref.com)

1. Given your circle, draw a line through the center, C, and the point on the circle, P. This will give us the diameter of the circle.
2. Pick a random point, A, anywhere inside the circle that is not too close to the diameter.
3. Construct a circle with center A and radius equal to AP.
4. Mark the intersection of this circle and our diameter line. Mark this R.
5. Construct a line through R and A, and mark the intersection of this line with the circle as point M.
6. Draw a line from M to P. This gives us our tangent line.

Another definition of a tangent, especially when it comes to a line tangent to a circle, is that the tangent line is perpendicular to the radius of the circle at that point. We can see this is true by the construction we just did above.

Now, we can move on to more difficult tasks: making a circle tangent to two other circles!!!

The first thing we need to realize is that there are three main different ways to draw two circles. We can have two circles intersecting each other, two circles that are completely apart, and one circle inside the other circle. In each of these different ways, there are also two different tangent circles that we can come up with. Thus, The rest of this investigation will be looking at how we construct each of these, as well as their locus, as that could provide for some more interesting exploration.

Two Intersecting Circles

Let's look at step by step instructions on how to construct the tangent circles in this case. First, we'll look at how to construct the tangent circle such that it is "inside" one of the circles.

1. Draw your two circles. We will label these the Outside circle and the Inside circle.
2. Pick a point on the outside circle (label it point P).
3. Draw a line through this point and the center of the circle.
4. Use the radius of our inside circle to construct a new circle centered at point P.
5. Construct a segment from the center of our inside circle to the outer intersection
point of our new circle and the line we constructed in step 3.
6. Find the midpoint of this line and construct a perpendicular line at this point.
7. Mark the point of intersection of the perpendicular line and our original line (from step 3)
- this will be our center of the tangent circle.
8. Use this point (from step 7) and the original point P and construct a circle.

We can see that this circle constructed in step 8 is our tangent circle that is "inside" one of the two circles.

You can click here for the GSP tool and sketch for this construction.

Now, we will look at how to construct the tangent circle such that it is outside both circles.

1. Draw your two circles. We will label these the Outside circle and the Inside circle.
2. Pick a point on the outside circle (label it point P).
3. Draw a line through this point and the center of the circle.
4. Use the radius of our inside circle to construct a new circle centered at point P.
5. Construct a segment from the center of our inside circle to the inner intersection point
of our new circle and the line we constructed in step 3.
6. Fine the midpoint of this line and construct a perpendicular line at this point.
7. Mark the point of intersection of the perpendicular line and our original line (from step 3)
- this will be our center of the tangent circle.
8. Use this point (from step 7) and the original point P and construct a circle.

We can see that the circle we just constructed in step 8 is our tangent circle that is outside both circles.

You can click here for the GSP tool and sketch for this construction.

Two circles that are completely separate

Let's look at step by step instructions on how to construct the tangent circles in this case. First, we'll look at how to construct the tangent circle such that it is between each of the circles.

1. Draw your two circles. We will label these as lefty and righty respectively.
2. Pick a point, P, on righty.
3. Draw a line through P and the center of righty.
4. Using the radius of lefty, construct a circle that is centered at point P.
5. Mark the top intersection point of this circle and our line from step 3.
6. Construct a segment from this intersection point to the center of lefty.
7. Find the midpoint of this new line (from step 6) and construct a perpendicular
line through that point.
8. Mark the intersection of our perpendicular line and original line (from step 3).
This will be the center of our tangent circle.
9. Using the point from step 8 and our original point P, construct a circle. This gives us our tangent circle.

As we can see, the circle we constructed is tangent to both of the circles.

You can click here for the GSP tool and sketch for this construction.

Now, let's look at step by step instructions on how to construct the tangent circle so that it is outside each of our circles.

1. Draw your two circles. We will label these as lefty and righty respectively.
2. Pick a point, P, on righty.
3. Draw a line through P and the center of righty.
4. Using the radius of lefty, construct a circle that is centered at point P.
5. Mark the bottom intersection point of this circle and our line from step 3.
6. Construct a segment from this intersection point to the center of lefty.
7. Find the midpoint of this new line (from step 6) and construct a perpendicular line
through that point.
8. Mark the intersection of our perpendicular line and original line (from step 3).
This will be the center of our tangent circle.
9. Using the point from step 8 and our original point P, construct a circle. This gives us our tangent circle.

We can see that the circle we just constructed is tangent to both of our original circles on the OUTside.

You can click here for the GSP tool and sketch of this construction.

One circle inside the second circle

Let's look at step by step instructions on how to construct the tangent circles in this case. First, we'll look at how to construct the tangent circle such that it is "between" our two circles.

1. Draw your two circles such that one lies inside the other. We will name our circles Outy and Inny respectively.
2. Pick a point, P, on Outy.
3. Draw a line through the center of Outy that goes through P.
4. Using this new point as the center, construct a circle with a radius
equal to the radius of Inny.
5. Mark the outer intersection point of our circle and the line we drew in step 3.
6. Draw a line between our intersection point and the center of Inny.
7. Find the midpoint of this line.
8. Construct a perpendicular line at this point.
9. The intersection of our perpendicular line and our line from step 3 will give us the
center of our tangent circle.
10. Using this center and our original point P, construct a circle. This gives us our tangent circle.

We can see that we have constructed a circle that is tangent to point given circles and lies between the two.

You can click here for the GSP tool and sketch of this construction.

Now, let's look at how to construct the tangent circle such that it is outside our inner circle and inside our outer circle.

1. Draw your two circles such that one lies inside the other. We will name our circles Outy and Inny respectively.
2. Pick a point, P, on Outy.
3. Draw a line through the center of Outy that goes through P.
4. Using this new point as the center, construct a circle with a radius
equal to the radius of Inny.
5. Mark the inner intersection point of our circle and the line we drew in step 3.
6. Draw a line between our intersection point and the center of Inny.
7. Find the midpoint of this line.
8. Construct a perpendicular line at this point.
9. The intersection of our perpendicular line and our line from step 3 will give us the center of our tangent circle.
10. Using this center and our original point P, construct a circle. This gives us our tangent circle.

We can see that this construction gives us a circle that is tangent to both of our given circles and lies outside our smaller circle.

You can click here for the GSP tool and sketch of the above construction.

What about the Locus of each of these tangents?

 Type of Circle Tangent Location Picture of Tangent Locus Description of Locus One circle inside the other circle. Between the inner circle and outer circle Ellipse One circle inside the other circle Outside the inner circle while still inside the outer circle Ellipse Two circles intersecting each other Inside one of the circles Ellipse Two circles intersecting each other Around both circles Hyperbola Completely detached Around one and between the other one Hyperbola Completely detached Around both Hyperbola