Draw this parabola and triangle in coordinate plane in  your copy.

To find the length of the side of the equilateral triangle inscribed in the parabola y^2 = 4ax, we need to find the coordinates of the three vertices of the triangle.

The vertex of the parabola is at the origin (0, 0). Let's assume one vertex of the equilateral triangle is at point A(a, 2a), where the distance from the origin to A is a.

Since the triangle is equilateral, the other two vertices B and C are obtained by rotating point A by 120 degrees and 240 degrees counterclockwise, respectively.

To rotate a point (x, y) by an angle of theta counterclockwise, the new coordinates (x', y') can be found using the following formulas:

x' = x * cos(theta) - y * sin(theta)
y' = x * sin(theta) + y * cos(theta)

Let's find the coordinates of points B and C by rotating point A by 120 degrees and 240 degrees counterclockwise, respectively.

For point B:
x' = a * cos(120) - (2a) * sin(120)
= -a/2 - a * sqrt(3)/2
= -a(1 + sqrt(3))/2

y' = a * sin(120) + (2a) * cos(120)
= a * sqrt(3)/2 - 2a/2
= a(sqrt(3) - 1)/2

So, point B is (-a(1 + sqrt(3))/2, a(sqrt(3) - 1)/2).

For point C:
x' = a * cos(240) - (2a) * sin(240)
= -a/2 + a * sqrt(3)/2
= a(sqrt(3) - 1)/2

y' = a * sin(240) + (2a) * cos(240)
= -a * sqrt(3)/2 - 2a/2
= -a(1 + sqrt(3))/2

So, point C is (a(sqrt(3) - 1)/2, -a(1 + sqrt(3))/2).

Now, we can find the length of the side of the equilateral triangle by calculating the distance between points A and B (or A and C).

Using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Distance = sqrt((-a(1 + sqrt(3))/2 - a)^2 + (a(sqrt(3) - 1)/2 - 2a)^2)
= sqrt(a^2(1 + sqrt(3))^2/4 + a^2(sqrt(3) - 1)^2/4)
= sqrt(a^2(1 + 2sqrt(3) + 3) + a^2(3 - 2sqrt(3) + 1)/4)
= sqrt(a^2(4 + 4sqrt(3))/4)
= sqrt(a^2(1 + sqrt(3)))

Therefore, the length of the side of the equilateral triangle inscribed in the parabola y^2 = 4ax is sqrt(a^2(1 + sqrt(3))