Find the area of an equilateral triangle in which length of perpendicular on sides from a point D inside the `Delta ` are `2 sqrt(3) + 3 sqrt(3) and 5 sqrt 3 `

Answer

To find the area of the equilateral triangle given the lengths of the perpendiculars from a point inside the triangle, we can follow these steps:

### Step 1: Understand the Problem
We have an equilateral triangle \( ABC \) and a point \( D \) inside it. The lengths of the perpendiculars from point \( D \) to the sides \( AB \), \( BC \), and \( CA \) are given as \( h_1 = 2\sqrt{3} \), \( h_2 = 3\sqrt{3} \), and \( h_3 = 5\sqrt{3} \).

### Step 2: Use the Area Formula
The area \( A \) of triangle \( ABC \) can be expressed in terms of the side length \( a \) of the equilateral triangle and the heights from point \( D \) to the sides:
\[ A = \frac{1}{2} \times a \times h_1 + \frac{1}{2} \times a \times h_2 + \frac{1}{2} \times a \times h_3 \]
This can be simplified to:
\[ A = \frac{a}{2} (h_1 + h_2 + h_3) \]

### Step 3: Substitute the Values
Now, substitute the values of the heights:
\[ A = \frac{a}{2} (2\sqrt{3} + 3\sqrt{3} + 5\sqrt{3}) \]
Calculating the sum of the heights:
\[ h_1 + h_2 + h_3 = 2\sqrt{3} + 3\sqrt{3} + 5\sqrt{3} = (2 + 3 + 5)\sqrt{3} = 10\sqrt{3} \]
So, we have:
\[ A = \frac{a}{2} \times 10\sqrt{3} = 5a\sqrt{3} \]

### Step 4: Relate the Area to the Side Length
The area of an equilateral triangle can also be expressed as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

### Step 5: Set the Two Area Expressions Equal
Now, we can set the two expressions for the area equal to each other:
\[ 5a\sqrt{3} = \frac{\sqrt{3}}{4} a^2 \]

### Step 6: Solve for \( a \)
To eliminate \( \sqrt{3} \) from both sides, we divide by \( \sqrt{3} \):
\[ 5a = \frac{1}{4} a^2 \]
Multiplying both sides by 4 gives:
\[ 20a = a^2 \]
Rearranging gives:
\[ a^2 - 20a = 0 \]
Factoring out \( a \):
\[ a(a - 20) = 0 \]
Thus, \( a = 0 \) or \( a = 20 \). Since \( a \) represents the side length of the triangle, we take \( a = 20 \).

### Step 7: Calculate the Area
Now we can find the area of the equilateral triangle:
\[ A = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} (20^2) = \frac{\sqrt{3}}{4} \times 400 = 100\sqrt{3} \]

### Final Answer
The area of the equilateral triangle is:
\[ \boxed{100\sqrt{3}} \]