The area of the triangle formed by three points on a parabola is how many times the area of the triangle formed by the tangents at these points?

Answer

To solve the problem of finding how many times the area of the triangle formed by three points on a parabola is compared to the area of the triangle formed by the tangents at these points, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points on the Parabola:**
Let the three points on the parabola \( y^2 = 4ax \) be:
- \( A(t_1) = (at_1^2, 2at_1) \)
- \( B(t_2) = (at_2^2, 2at_2) \)
- \( C(t_3) = (at_3^2, 2at_3) \)

2. **Calculate the Area of Triangle Formed by the Points:**
The area \( A \) of the triangle formed by points \( A \), \( B \), and \( C \) can be calculated using the determinant formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ \text{Area} = \frac{1}{2} \left| at_1^2(2at_2 - 2at_3) + at_2^2(2at_3 - 2at_1) + at_3^2(2at_1 - 2at_2) \right| \]
Simplifying this expression, we get:
\[ \text{Area} = \frac{a^2}{2} \left| (t_1^2(t_2 - t_3) + t_2^2(t_3 - t_1) + t_3^2(t_1 - t_2)) \right| \]

3. **Find the Tangents at the Points:**
The equation of the tangent to the parabola at point \( (at_i^2, 2at_i) \) is given by:
\[ y = mx + \frac{a}{m} \]
where \( m = \frac{2a}{t_i} \).

4. **Calculate the Intersection of the Tangents:**
The intersection points of the tangents at points \( A(t_1) \), \( B(t_2) \), and \( C(t_3) \) can be calculated. The coordinates of the intersection points can be derived from the tangent equations.

5. **Calculate the Area of the Triangle Formed by the Tangents:**
Using the same determinant formula for the area of the triangle formed by the intersection points of the tangents, we can derive the area \( A_T \).

6. **Relate the Areas:**
After calculating both areas, we can compare them:
\[ \text{Area of triangle from points} = 2 \times \text{Area of triangle from tangents} \]
Therefore, the area of the triangle formed by the points on the parabola is double the area of the triangle formed by the tangents at these points.

### Final Answer:
The area of the triangle formed by the three points on the parabola is **2 times** the area of the triangle formed by the tangents at these points.

---