Suppose that the circles in the figure below are tangent to one another. The radius of the circle centered at is one unit, and the radius of the circle centered at is two units. What are the radii of the congruent circles centered at and ?

Source: NCTM Problem to Ponder May 2011

**SOLUTION
**Since the two circles and are congruent, the centers of circles are collinear. The radius of circle equals 3. Let be the radius of circle .

Distance

Distance

Distance

Multiply Eq. (1) by and add to Eq. (2)

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Multiply Eq. (2) by and add to Eq. (3)

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Multiply Eq. (5) by and add to Eq. (4)

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**Answer**: