Mathematical Practices
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Tangent circles are coplanar circles that intersect in exactly one point.
See solution.
Circles ⊙ A, ⊙ B, and ⊙ C consist of points that are 3 units from the centers. We are asked to draw the three circles so that each is tangent to the other two. Recall that tangent circles are coplanar circles that intersect at exactly one point. Let's start by drawing ⊙ A and ⊙ B so that they are tangent to each other.
Finally, we can draw ⊙ C.
Next we will draw a larger circle ⊙ D that is tangent to the other three circles.
We can see that point D lies in the space between the circles ⊙ A, ⊙ B, and ⊙ C. To determine whether the distance from D to a point on ⊙ D is less than, greater than, or equal to 6, let's first draw a line segment from D through A, to a point on ⊙ D.
The distance from one point of intersection of the segment with ⊙ A to the second point of intersection is 6 units. The point of intersection of the segment with ⊙ D is also a point of intersection of the segment with ⊙ A. Since point D is outside ⊙ A, its distance to a point on ⊙ D is greater than 6.