Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true:
$[DEFGHI]\geq k\cdot [ABC]$
Note: $[X]$ denotes the area of polygon $X$. 
1. **Understanding the problem**: We need to find the greatest positive real constant \( k \) such that for any triangle \( \triangle ABC \), the inequality \( [DEFGHI] \geq k \cdot [ABC] \) holds. Here, \( [X] \) denotes the area of polygon \( X \).

2. **Analyzing the given configuration**: We have squares \( BADE \), \( CBFG \), and \( ACHI \) drawn externally on the sides of \( \triangle ABC \). We need to find the area of the hexagon \( DEFGHI \).

3. **Area relations**: By the properties of the squares and the triangle, we can see that:
   \[
   [DAI] = [HCG] = [EBF] = [ABC]
   \]
   This is because each of these triangles shares a side with \( \triangle ABC \) and has the same height relative to that side.

4. **Total area of the hexagon**: The area of the hexagon \( DEFGHI \) can be expressed as the sum of the areas of the three triangles \( DAI \), \( HCG \), and \( EBF \), plus the area of \( \triangle ABC \):
   \[
   [DEFGHI] = [DAI] + [HCG] + [EBF] + [ABC] = 3[ABC] + [ABC] = 4[ABC]
   \]

5. **Formulating the inequality**: We need to find the maximum \( k \) such that:
   \[
   4[ABC] \geq k \cdot [ABC]
   \]
   Simplifying this, we get:
   \[
   4 \geq k
   \]

6. **Conclusion**: The greatest positive real constant \( k \) that satisfies the inequality for any triangle \( \triangle ABC \) is \( k = 4 \).

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