If we create triangle ABC as pictured below, we can see that when angle ABC is acute, then the tangent circle will surround circle c1. When angle ABC is obtuse, the tangent circle will surround circle c2.

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Here is a GSP animation to illustrate. Double click on the animation button in order to begin the animation.

What happens to the tangent circle when angle ABC is a right angle? What is the radius of the tangent circle?