The joint equation of two altitudes of an equilateral triangle is `(sqrt(3)x-y+8-4sqrt(3)) (-sqrt(3)x-y+12 +4sqrt(3)) = 0` The third altitude has the equation

Explanation

To find the equation of the third altitude of the equilateral triangle given the joint equation of two altitudes, we can follow these steps:

### Step 1: Identify the equations of the two given altitudes
The joint equation provided is:
\[ (\sqrt{3}x - y + 8 - 4\sqrt{3})(-\sqrt{3}x - y + 12 + 4\sqrt{3}) = 0 \]
From this, we can extract the two altitude equations:
1. \(\sqrt{3}x - y + 8 - 4\sqrt{3} = 0\)
2. \(-\sqrt{3}x - y + 12 + 4\sqrt{3} = 0\)

### Step 2: Rewrite the equations in standard form
The first altitude can be rewritten as:
\[ \sqrt{3}x - y + (8 - 4\sqrt{3}) = 0 \]
The second altitude can be rewritten as:
\[ -\sqrt{3}x - y + (12 + 4\sqrt{3}) = 0 \]

### Step 3: Identify coefficients
From the equations, we identify:
- For the first altitude: \(a_1 = \sqrt{3}, b_1 = -1, c_1 = 8 - 4\sqrt{3}\)
- For the second altitude: \(a_2 = -\sqrt{3}, b_2 = -1, c_2 = 12 + 4\sqrt{3}\)

### Step 4: Check the condition for the third altitude
The condition for the third altitude is given by:
\[ a_1 a_2 + b_1 b_2 < 0 \]
Calculating this:
\[ a_1 a_2 = \sqrt{3} \cdot (-\sqrt{3}) = -3 \]
\[ b_1 b_2 = (-1)(-1) = 1 \]
Thus,
\[ -3 + 1 = -2 < 0 \]
This condition holds true, confirming that the third altitude exists.

### Step 5: Find the equation of the third altitude
The equation of the third altitude can be derived from the relationship:
\[ \text{Third Altitude: } \sqrt{3}x - y + (8 - 4\sqrt{3}) = -(-\sqrt{3}x - y + (12 + 4\sqrt{3})) \]
This simplifies to:
\[ \sqrt{3}x - y + 8 - 4\sqrt{3} = \sqrt{3}x + y - 12 - 4\sqrt{3} \]
Rearranging gives:
\[ 2y = 20 \quad \Rightarrow \quad y - 10 = 0 \]

### Final Answer
The equation of the third altitude is:
\[ y - 10 = 0 \]