`A, B, C` are three points on the rectangular hyperbola `xy = c^2`, The area of the triangle formed by the points `A, B` and `C` is

Answer

To find the area of the triangle formed by the points A, B, and C on the rectangular hyperbola \( xy = c^2 \), we can follow these steps:

### Step 1: Identify the points
Let the points \( A, B, C \) be given as:
- \( A(t_1) = (t_1, \frac{c^2}{t_1}) \)
- \( B(t_2) = (t_2, \frac{c^2}{t_2}) \)
- \( C(t_3) = (t_3, \frac{c^2}{t_3}) \)

### Step 2: Use the formula for the area of a triangle
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the determinant formula:
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| \]

### Step 3: Substitute the points into the determinant
Substituting the coordinates of points A, B, and C into the formula:
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} t_1 & \frac{c^2}{t_1} & 1 \\ t_2 & \frac{c^2}{t_2} & 1 \\ t_3 & \frac{c^2}{t_3} & 1 \end{vmatrix} \right| \]

### Step 4: Calculate the determinant
Calculating the determinant:
\[ = \frac{1}{2} \left( t_1 \left( \frac{c^2}{t_2} - \frac{c^2}{t_3} \right) - \frac{c^2}{t_1} \left( t_2 - t_3 \right) + t_2 \left( \frac{c^2}{t_3} - \frac{c^2}{t_1} \right) \right) \]

This simplifies to:
\[ = \frac{1}{2} \left( \frac{c^2}{t_1} (t_3 - t_2) + \frac{c^2}{t_2} (t_1 - t_3) + \frac{c^2}{t_3} (t_2 - t_1) \right) \]

### Step 5: Factor out common terms
Factoring out \( c^2 \) from the expression:
\[ = \frac{c^2}{2} \left( \frac{(t_3 - t_2)}{t_1} + \frac{(t_1 - t_3)}{t_2} + \frac{(t_2 - t_1)}{t_3} \right) \]

### Step 6: Final expression for the area
The final area of triangle ABC can be expressed as:
\[ \text{Area} = \frac{c^2}{2 t_1 t_2 t_3} (t_1 - t_2)(t_2 - t_3)(t_3 - t_1) \]

### Conclusion
Thus, the area of triangle ABC formed by the points on the hyperbola \( xy = c^2 \) is:
\[ \text{Area} = \frac{c^2}{2 t_1 t_2 t_3} (t_1 - t_2)(t_2 - t_3)(t_3 - t_1) \]