Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally.  The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$
1. **Identify the centers and radii of the circles:**
   Let \( \gamma_1, \gamma_2, \gamma_3 \) be three circles of unit radius that touch each other externally. Let \( O_1, O_2, O_3 \) be the centers of these circles. Since the circles touch each other externally, the distance between any two centers is \( 2 \) units (sum of the radii).

2. **Form the triangle \( \triangle O_1O_2O_3 \):**
   The triangle \( \triangle O_1O_2O_3 \) is an equilateral triangle with each side equal to \( 2 \) units.

3. **Determine the homothety ratio:**
   The triangle \( \triangle XYZ \) formed by the common tangents to each pair of circles is homothetic to \( \triangle O_1O_2O_3 \). The homothety ratio can be determined by considering the distances from the center of homothety \( O \) (which is the circumcenter of both triangles) to the sides of the triangles.

4. **Calculate the distance from \( O \) to the sides of \( \triangle O_1O_2O_3 \):**
   The distance from \( O \) to a side of \( \triangle O_1O_2O_3 \) is the altitude of the equilateral triangle divided by \( \sqrt{3} \). The altitude of an equilateral triangle with side length \( 2 \) is \( \sqrt{3} \), so the distance is:
   \[
   \frac{\sqrt{3}}{\sqrt{3}} = 1
   \]

5. **Calculate the distance from \( O \) to the sides of \( \triangle XYZ \):**
   The distance from \( O \) to a side of \( \triangle XYZ \) is the distance from \( O \) to a side of \( \triangle O_1O_2O_3 \) plus the radius of the circles, which is \( 1 \). Therefore, the distance is:
   \[
   1 + 1 = 2
   \]

6. **Determine the homothety ratio:**
   The homothety ratio is the ratio of the distances from \( O \) to the sides of \( \triangle XYZ \) and \( \triangle O_1O_2O_3 \):
   \[
   \frac{2}{1} = 2
   \]

7. **Calculate the side length of \( \triangle XYZ \):**
   Since \( \triangle XYZ \) is homothetic to \( \triangle O_1O_2O_3 \) with a ratio of \( 2 \), the side length of \( \triangle XYZ \) is:
   \[
   2 \times 2 = 4
   \]

The final answer is \( \boxed{4} \).