Example 10. Two vertices of an equilateral triangle are on the parabola $\mathrm{y}^{2}=2 \mathrm{px}$, and the other vertex is at the origin. Find the side length of this triangle.
To find the length of OA as shown in Figure 7, it is sufficient to determine the coordinates of point A, denoted as \(\left(x_{1}, y_{1}\right)\).

Since A is the intersection point of the parabola and the line OA, and the equation of the line OA is
\[ y = \tan 30^{\circ} \cdot x = \frac{\sqrt{3}}{3} x, \]

\(\left(x_{1}, y_{1}\right)\) is the solution to the system of equations
\[ \left\{\begin{array}{l}
y = \frac{\sqrt{3}}{3} x, \\
y^{2} = 2 p x
\end{array}\right. \]

Solving this, we get \(x_{1} = 6 p, y_{1} = 2 \sqrt{3} p\).
Therefore, \(|OA| = \sqrt{x_{1}^2 + y_{1}^2} = \sqrt{(6 p)^2 + (2 \sqrt{3} p)^2} = 4 \sqrt{3} p\), which means the side length of the equilateral triangle is \(4 \sqrt{3} p\).