Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$ and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$.
[img]https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png[/img]
1. **Calculate the area of the equilateral triangle \( \triangle ABC \) with side length \( M \):**

   The area \( A \) of an equilateral triangle with side length \( M \) is given by:
   \[
   A = \frac{M^2 \sqrt{3}}{4}
   \]

2. **Identify the smaller triangles and their areas:**

   Given that \( m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2} \), each smaller triangle \( \triangle AE_1G_2 \), \( \triangle BF_1E_2 \), and \( \triangle CG_1F_2 \) is also an equilateral triangle with side length \( m \).

   The area of each smaller equilateral triangle with side length \( m \) is:
   \[
   A_{\text{small}} = \frac{m^2 \sqrt{3}}{4}
   \]

3. **Calculate the combined area of the three smaller triangles:**

   Since there are three such smaller triangles, their combined area is:
   \[
   3 \times \frac{m^2 \sqrt{3}}{4} = \frac{3m^2 \sqrt{3}}{4}
   \]

4. **Express the area of the hexagon \( E_1E_2F_1F_2G_1G_2 \):**

   The area of the hexagon is the area of the large equilateral triangle minus the combined area of the three smaller triangles:
   \[
   A_{\text{hexagon}} = \frac{M^2 \sqrt{3}}{4} - \frac{3m^2 \sqrt{3}}{4}
   \]

5. **Set up the equation based on the given condition:**

   According to the problem, the area of the hexagon equals the combined areas of the three smaller triangles:
   \[
   \frac{M^2 \sqrt{3}}{4} - \frac{3m^2 \sqrt{3}}{4} = \frac{3m^2 \sqrt{3}}{4}
   \]

6. **Solve for \( M^2 \) in terms of \( m^2 \):**

   Simplify the equation:
   \[
   \frac{M^2 \sqrt{3}}{4} - \frac{3m^2 \sqrt{3}}{4} = \frac{3m^2 \sqrt{3}}{4}
   \]
   \[
   \frac{M^2 \sqrt{3}}{4} = \frac{6m^2 \sqrt{3}}{4}
   \]
   \[
   M^2 = 6m^2
   \]

7. **Find the ratio \( \frac{m}{M} \):**

   Taking the square root of both sides:
   \[
   M = \sqrt{6} m
   \]
   \[
   \frac{m}{M} = \frac{1}{\sqrt{6}}
   \]

The final answer is \( \boxed{\frac{1}{\sqrt{6}}} \).