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The problem involves an equilateral triangle \( \triangle ABC \) with side length 4, where the sides \( AB \) and \( BC \) are tangents to the circle at points \( E \) and \( F \) respectively. We are tasked with finding the radius \( r \) of the circle \( O \).

### Solution Outline:

1. **Equilateral Triangle Properties**:
   - Since \( \triangle ABC \) is equilateral, all sides are equal (4 units each), and all internal angles are 60°.

2. **Tangents to a Circle**:
   - The tangents from a point to a circle are equal in length. Therefore, \( AE = AF \) and \( BE = BF \).
   - The center \( O \) of the circle lies on the angle bisector of \( \angle ACB \), which also acts as the altitude from \( C \) to \( AB \) in \( \triangle ABC \).

3. **Altitude of the Equilateral Triangle**:
   - The altitude of an equilateral triangle can be calculated as \( h = \frac{\sqrt{3}}{2} \times \text{side length} \).
   - Here, the altitude from \( C \) to \( AB \) is \( h = \frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3} \).

4. **Geometry and Relation to Circle**:
   - The center of the circle \( O \) is the incenter of \( \triangle ABC \) (since the tangents are on two sides of the triangle).
   - The inradius (the radius of the circle inscribed in \( \triangle ABC \)) can be found using the formula for the inradius \( r \) of an equilateral triangle:
     \[
     r = \frac{s \sqrt{3}}{6}
     \]
     where \( s \) is the side length of the triangle.

5. **Calculating the Radius**:
   - Substituting \( s = 4 \) into the formula:
     \[
     r = \frac{4 \sqrt{3}}{6} = \frac{2 \sqrt{3}}{3}
     \]

Thus, the radius \( r \) of the circle is \( \frac{2 \sqrt{3}}{3} \).

Would you like further details or clarification? Here are five related questions that expand on this:

1. How do we derive the formula for the inradius of an equilateral triangle?
2. Can we calculate the area of the equilateral triangle using its inradius?
3. How would the result change if the triangle were not equilateral but isosceles?
4. What is the relationship between the circumradius and inradius of an equilateral triangle?
5. How does the position of the circle change if one of the triangle's angles changes?

**Tip**: When working with equilateral triangles, their symmetry often allows us to apply simplified geometric properties and formulas.

As shown in the diagram, △ABC is an equilateral triangle with a side length of 4. AB and BC are tangents of circle O at points E and F. Find the radius of circle O.

Learn how to calculate the radius of a circle inscribed in an equilateral triangle with side length 4. This problem involves key geometric concepts like tangents, inradius, and triangle properties. Ideal for students in Grades 9-12.

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