Megan Langford

What does it mean to be tangent? Well, the definition of tangent is a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point.

So what about a tangent circle? A circle is tangent to 2 other circles if it touches each circle at exactly one point and does not intersect them. What would this look like? First, let’s explore 2 different scenarios where we will construct the tangent circle to both original circles. In the first example, we will place one of the circles inside of the larger one, as pictured.

In our second example, let’s place the smaller circle outside of the larger one, like this one.

We will now construct circles that are tangent to each of the originals for both of these examples. Hence, our final result should appear as the red tangent circle in the following 2 images.

And for the example where the circles are separated, our tangent circle will appear as follows.

Constructing Tangent Circles

So now let’s take the first example where the smaller circle sits inside the larger one. The first thing we need to do is to construct a line from the midpoint of the larger circle and passing through it.

Now, we should mark the point where the line intersects with the edge of the circle. Then construct a circle identical to the smaller one centered at this intersection point.

Next, we should identify the point of intersection between the line and the outer edge of the new circle. Then we should construct a line between this point and the center of the original small circle.

Then, construct the midpoint of this line and then a perpendicular line through the midpoint.

Since our smaller circles are congruent, we know that the pink area below represents two congruent isosceles triangles.

So we can now construct our tangent circle. The center is located where the line through the midpoint of the larger circle and the perpendicular line we constructed intersect. We will of course have the circle be of the exact size to touch each of the original circles in exactly one point and not intersect.

As we can clearly see, this is the case from the red circle that we constructed.

Here is a link for the construction if you would like to explore its other possibilities.

Now, what would we change about this construction if the smaller circle were instead positioned outside the larger one, similar to our second example?

First, we would create a line through the midpoint of the larger circle.

We will now replicate the smaller circle onto the closest point to it using the line intersect with the larger circle as its center.

Next, we will draw a line connecting the radius of the original small circle with the intersection point between the line and the closest point on the newer circle towards the center of the larger one. Then we will construct a midpoint and a perpendicular line to this one.

Now, we will construct a line between the centers of the smaller and the larger original circles. Then also construct its midpoint and a perpendicular line there.

We are now ready to construct the tangent circle!