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This problem is related to circles inscribed within a right triangle, where the triangle has side lengths of \( AB = 10 \), \( BC = 8 \), and \( CA = 6 \). The task is to find the radii of the three inscribed circles, which fit between one leg of the triangle and the hypotenuse.

Given that these circles are tangent to both the sides of the triangle and to each other, this is a known geometric configuration where the radii of the circles decrease in size as they approach the smaller side of the triangle. The answer is given as \( \frac{10}{9} \), which likely represents the radius of one of the circles.

Let’s break down how to calculate the radius in such a scenario.

### Step-by-step approach:

1. **Pythagorean theorem**:
   Since \( \triangle ABC \) is a right triangle, we can verify the side lengths:
   \[
   AB^2 = BC^2 + CA^2
   \]
   \[
   10^2 = 8^2 + 6^2
   \]
   \[
   100 = 64 + 36
   \]
   \[
   100 = 100
   \]
   This confirms that it is a right triangle.

2. **Area of the triangle**:
   The area \( A \) of the right triangle can be calculated using the formula:
   \[
   A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 6 = 24
   \]

3. **Semiperimeter of the triangle**:
   The semiperimeter \( s \) is given by:
   \[
   s = \frac{AB + BC + CA}{2} = \frac{10 + 8 + 6}{2} = 12
   \]

4. **Inradius of the entire triangle**:
   The inradius \( r \) of the triangle can be found using the formula:
   \[
   r = \frac{A}{s} = \frac{24}{12} = 2
   \]
   This gives the radius of the incircle of the whole triangle.

5. **For the smaller circles**:
   The three smaller circles are inscribed within the remaining space of the triangle, and their radii must be calculated based on the tangential constraints with respect to the sides and each other.

The provided solution gives the radius of the smaller circles as \( \frac{10}{9} \).

Would you like further details on this solution, or do you have additional questions?

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Here are five relative questions:

1. How do you derive the formula for the radius of smaller inscribed circles in general?
2. What is the general relationship between the radius of the incircle and the triangle's sides?
3. How can the Descartes Circle Theorem be applied in such configurations?
4. Can this approach be generalized for any right triangle, or only for specific ones?
5. How would the solution change if the triangle were not right-angled?

**Tip**: Whenever solving problems involving inscribed circles, it's helpful to use properties like tangency and semiperimeter to simplify complex relationships.

In a right triangle ABC with sides AB = 10, BC = 8, and CA = 6, three circles are inscribed, each tangent to two sides and to each other. Find the radius of these circles.

Solve the problem of finding the radius of inscribed circles within a right triangle with sides 10, 8, and 6. Learn about the Pythagorean theorem, area calculation, and inradius formula. Suitable for Grades 10-12.

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