Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim  \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
(Here $[P]$ denotes the area of polygon $P$.)
1. **Understanding the Problem:**
   We are given an equilateral triangle \( \triangle ABC \) with points \( D, E, F \) on sides \( AB, BC, \) and \( CA \) respectively such that \( [ADF] = [BED] + [CEF] \) and \( \triangle ADF \sim \triangle BED \sim \triangle CEF \). We need to find the ratio \( \frac{[ABC]}{[DEF]} \) in the form \( \frac{a+b\sqrt{c}}{d} \) and then determine \( a + b + c + d \).

2. **Using Similarity:**
   Since \( \triangle ADF \sim \triangle BED \sim \triangle CEF \), the ratios of corresponding sides are equal. Let the side length of \( \triangle ABC \) be \( s \). Let the side lengths of \( \triangle DEF \) be \( x, y, z \) respectively.

3. **Area Relationships:**
   The area of \( \triangle ABC \) is:
   \[
   [ABC] = \frac{\sqrt{3}}{4} s^2
   \]
   Since \( \triangle ADF \sim \triangle BED \sim \triangle CEF \), the areas of these triangles are proportional to the square of the sides. Let \( k \) be the ratio of similarity.

4. **Using the Given Area Condition:**
   Given \( [ADF] = [BED] + [CEF] \), and using the similarity ratio \( k \):
   \[
   [ADF] = k^2 [ABC]
   \]
   \[
   [BED] = k^2 [ABC]
   \]
   \[
   [CEF] = k^2 [ABC]
   \]
   Therefore:
   \[
   k^2 [ABC] = k^2 [ABC] + k^2 [ABC]
   \]
   This implies:
   \[
   k^2 = \frac{1}{2}
   \]

5. **Finding the Ratio \( \frac{[ABC]}{[DEF]} \):**
   The area of \( \triangle DEF \) is:
   \[
   [DEF] = k^2 [ABC] = \frac{1}{2} [ABC]
   \]
   Therefore:
   \[
   \frac{[ABC]}{[DEF]} = \frac{[ABC]}{\frac{1}{2} [ABC]} = 2
   \]

6. **Expressing the Ratio in the Given Form:**
   The ratio \( \frac{[ABC]}{[DEF]} \) can be expressed as \( \frac{2+0\sqrt{3}}{1} \), where \( a = 2 \), \( b = 0 \), \( c = 3 \), and \( d = 1 \).

7. **Summing the Values:**
   \[
   a + b + c + d = 2 + 0 + 3 + 1 = 6
   \]

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