If `(x_i,y_i),i=1,2,3` are the vertices of an equilateral triangle such that `(x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2`, then find the value of `(x_1+x_2+x_3)/(y_1+y_2+y_3)`

Explanation

To solve the problem, we need to analyze the given condition and find the required value step by step.

### Step 1: Understand the given condition
We are given that the vertices of an equilateral triangle are \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) such that:
\[ (x_1 + 2)^2 + (y_1 - 3)^2 = (x_2 + 2)^2 + (y_2 - 3)^2 = (x_3 + 2)^2 + (y_3 - 3)^2 \]
This means that the distances from the point \((-2, 3)\) to each of the triangle's vertices are equal.

### Step 2: Identify the center of the triangle
The point \((-2, 3)\) acts as the circumcenter of the equilateral triangle. In an equilateral triangle, the circumcenter coincides with the centroid.

### Step 3: Use the properties of the centroid
The centroid \((G)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]
Since the circumcenter and centroid coincide, we have:
\[ \frac{x_1 + x_2 + x_3}{3} = -2 \quad \text{and} \quad \frac{y_1 + y_2 + y_3}{3} = 3 \]

### Step 4: Solve for \(x_1 + x_2 + x_3\) and \(y_1 + y_2 + y_3\)
From the equations above, we can multiply both sides by 3 to find:
\[ x_1 + x_2 + x_3 = -6 \quad \text{and} \quad y_1 + y_2 + y_3 = 9 \]

### Step 5: Calculate the required ratio
Now we need to find the value of:
\[ \frac{x_1 + x_2 + x_3}{y_1 + y_2 + y_3} \]
Substituting the values we found:
\[ \frac{x_1 + x_2 + x_3}{y_1 + y_2 + y_3} = \frac{-6}{9} = -\frac{2}{3} \]

### Final Answer
Thus, the value of \(\frac{x_1 + x_2 + x_3}{y_1 + y_2 + y_3}\) is:
\[ \boxed{-\frac{2}{3}} \]