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We are given the equation of a parabola \( y^2 = 8x \), and an equilateral triangle is inscribed in the parabola with its apex at the top of the parabola. Our goal is to find the area of the triangle.

### Key Steps:
1. **Equation of the Parabola**: 
   The given parabola is \( y^2 = 8x \), which can be rewritten in vertex form as \( x = \frac{y^2}{8} \). This is a standard sideways parabola, with its vertex at the origin (0,0).

2. **Apex of the Triangle**: 
   The apex of the triangle is at the top of the parabola. For a parabola of the form \( y^2 = 8x \), the "top" refers to the vertex, which is at \( (0,0) \).

3. **Equilateral Triangle Inscribed in the Parabola**: 
   The triangle is symmetric, with its base aligned horizontally. The two other vertices are on the parabola. Let the coordinates of these vertices be \( (-a, y_1) \) and \( (a, y_1) \), where \( y_1 \) is the height from the x-axis.

4. **Using Geometry of the Triangle**: 
   For an equilateral triangle, the height is related to the base by the formula:
   \[
   \text{Height of an equilateral triangle} = \frac{\sqrt{3}}{2} \times \text{Base}
   \]
   The base of the triangle is \( 2a \), so the height of the triangle, which is also the y-coordinate of the third vertex, is \( \frac{\sqrt{3}}{2} \times 2a = \sqrt{3}a \).

5. **Parabola Constraint**: 
   The points \( (-a, y_1) \) and \( (a, y_1) \) lie on the parabola, so their coordinates must satisfy the equation of the parabola. For the point \( (a, y_1) \), we have:
   \[
   y_1^2 = 8a
   \]

6. **Substitute Height Relationship**: 
   The height of the triangle is also \( \sqrt{3}a \), so:
   \[
   \left( \sqrt{3}a \right)^2 = 8a
   \]
   Simplifying this equation:
   \[
   3a^2 = 8a
   \]
   Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
   \[
   3a = 8 \quad \Rightarrow \quad a = \frac{8}{3}
   \]

7. **Area of the Triangle**: 
   The area \( A \) of an equilateral triangle is given by:
   \[
   A = \frac{1}{2} \times \text{Base} \times \text{Height}
   \]
   The base is \( 2a = \frac{16}{3} \), and the height is \( \sqrt{3}a = \frac{8\sqrt{3}}{3} \). Therefore, the area is:
   \[
   A = \frac{1}{2} \times \frac{16}{3} \times \frac{8\sqrt{3}}{3}
   \]
   Simplifying:
   \[
   A = \frac{1}{2} \times \frac{128\sqrt{3}}{9} = \frac{64\sqrt{3}}{9}
   \]

Thus, the area of the equilateral triangle inscribed in the parabola is \( \frac{64\sqrt{3}}{9} \).

Would you like further details or clarifications?

Here are some related questions you might want to explore:
1. How do you derive the formula for the area of an equilateral triangle?
2. Can you find the length of each side of the triangle in this case?
3. How would the solution change if the parabola was vertical?
4. What if the triangle was not equilateral but isosceles? How would you approach it?
5. How does the height-to-base relationship of an equilateral triangle affect the geometry of inscribing it in other curves?

**Tip**: When working with symmetrical shapes like equilateral triangles, symmetry simplifies calculations, especially with parabolas or other even functions.

An equilateral triangle is drawn inside the parabola y^2 = 8x with its apex at the top of the parabola. Find the area of the triangle.

This problem explores how to calculate the area of an equilateral triangle inscribed in the parabola y^2 = 8x, with its apex at the top. Step-by-step instructions include geometry and algebraic methods. Suitable for high school students (Grade 10-12).

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