The vertex of an equilateral triangle is `(2,3)` and the equation of the opposite side is `x+y=2`. Then, the other two sides are `y-3=(2pmsqrt3)(x-2)`.

Explanation

To solve the problem step by step, we will find the equations of the other two sides of the equilateral triangle given one vertex and the equation of the opposite side.

### Step 1: Understand the Given Information
We have:
- Vertex of the triangle: \( A(2, 3) \)
- Equation of the opposite side: \( x + y = 2 \)

### Step 2: Rewrite the Equation of the Opposite Side
We can rewrite the equation \( x + y = 2 \) in slope-intercept form:
\[ y = -x + 2 \]
From this, we can identify the slope \( m_1 \) of the line:
\[ m_1 = -1 \]

### Step 3: Determine the Angle Between the Lines
In an equilateral triangle, the angle between any two sides is \( 60^\circ \). Therefore, we need to find the slopes \( m \) of the other two sides.

### Step 4: Use the Formula for the Angle Between Two Lines
The formula for the tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting \( \theta = 60^\circ \) (where \( \tan(60^\circ) = \sqrt{3} \)):
\[ \sqrt{3} = \left| \frac{-1 - m}{1 - m} \right| \]

### Step 5: Set Up the Equation
We can set up two equations based on the absolute value:
1. \( \sqrt{3} = \frac{-1 - m}{1 - m} \)
2. \( \sqrt{3} = \frac{1 + m}{1 - m} \)

### Step 6: Solve the First Equation
From the first equation:
\[ \sqrt{3}(1 - m) = -1 - m \]
Expanding and rearranging gives:
\[ \sqrt{3} - \sqrt{3}m = -1 - m \]
\[ \sqrt{3} + 1 = m + \sqrt{3}m \]
\[ m(1 + \sqrt{3}) = \sqrt{3} + 1 \]
\[ m = \frac{\sqrt{3} + 1}{1 + \sqrt{3}} \]

### Step 7: Solve the Second Equation
From the second equation:
\[ \sqrt{3}(1 - m) = 1 + m \]
Expanding gives:
\[ \sqrt{3} - \sqrt{3}m = 1 + m \]
Rearranging gives:
\[ \sqrt{3} - 1 = m + \sqrt{3}m \]
\[ m(1 + \sqrt{3}) = \sqrt{3} - 1 \]
\[ m = \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \]

### Step 8: Find the Slopes
After solving both equations, we find:
1. \( m_2 = 2 + \sqrt{3} \)
2. \( m_3 = 2 - \sqrt{3} \)

### Step 9: Write the Equations of the Other Two Sides
Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \), we can write the equations of the two sides:
1. For \( m_2 = 2 + \sqrt{3} \):
\[ y - 3 = (2 + \sqrt{3})(x - 2) \]
2. For \( m_3 = 2 - \sqrt{3} \):
\[ y - 3 = (2 - \sqrt{3})(x - 2) \]

### Final Result
Thus, the equations of the other two sides of the equilateral triangle are:
\[ y - 3 = (2 + \sqrt{3})(x - 2) \quad \text{and} \quad y - 3 = (2 - \sqrt{3})(x - 2) \]