Proof:

Given: The equation of the parabola is y^2 = 4ax.

Let's consider a point P(x1, y1) on the parabola.

Step 1: Finding the equation of the tangent at P(x1, y1)
The equation of a tangent to the parabola y^2 = 4ax at point P(x1, y1) is given by the slope-point form:

(y - y1) = m(x - x1), where m is the slope of the tangent.

Differentiating the equation of the parabola with respect to x, we get:
2yy' = 4a

Simplifying, we get:
y' = 2a/y

Substituting the coordinates of point P(x1, y1) in y', we get:
m = 2a/y1

So, the equation of the tangent becomes:
(y - y1) = (2a/y1)(x - x1)

Step 2: Finding the coordinates of the point of contact of the tangent with the x-axis
To find the x-coordinate of the point of contact, we set y = 0 in the equation of the tangent:
0 - y1 = (2a/y1)(x - x1)

Simplifying, we get:
x = x1 - (y1^2)/(2a)

Since the point lies on the x-axis, its y-coordinate is 0. Therefore, the coordinates of the point of contact are:
T(x1 - (y1^2)/(2a), 0)

Step 3: Finding the equation of the tangent at the vertex A
The vertex of the parabola y^2 = 4ax is (0, 0).

To find the equation of the tangent at the vertex, we can differentiate the equation of the parabola with respect to x and substitute x = 0.

Differentiating the equation of the parabola with respect to x, we get:
2yy' = 4a

Substituting x = 0, we get:
0 = 4a

So, the slope of the tangent at the vertex is 0, and the equation of the tangent becomes:
y - 0 = 0(x - 0)
y = 0

Therefore, the equation of the tangent at the vertex A is y = 0.

Step 4: Finding the coordinates of point Y
Since the equation of the tangent at the vertex is y = 0, the y-coordinate of point Y is 0. Therefore, the coordinates of point Y are (0, 0).

Step 5: Finding the coordinates of point G
Since TAYG is a rectangle, the x-coordinate of point G is the same as the y-coordinate of point Y, which is 0. Therefore, the coordinates of point G are (0, 0).

Step 6: Finding the locus of point G
The locus of point G is the set of all points that satisfy the condition that the x-coordinate is 0. In other words, the x-coordinate is always 0.

Therefore, the locus of point G is the line x = 0, which can be written as ax = 0.

H