Proof:

Let's consider a parabola with the equation y^2 = 4ax, where a is a constant and represents the focal length.

1. Deriving the Equation of the Normal Chord:
To find the equation of the normal chord to the parabola, we first need to find the slope of the tangent at a given point on the parabola.

Let's consider a point (h, k) on the parabola. The slope of the tangent at this point is given by dy/dx = -2ah.

Since the slope of the normal chord is the negative reciprocal of the slope of the tangent, the slope of the normal chord is 1/(2ah).

Now, let's find the equation of the normal chord passing through the point (h, k). Using the point-slope form, the equation of the normal chord is given by y - k = (1/(2ah))(x - h).

Simplifying this equation, we get 2ahy - 2ahk = x - h.

2. Finding the Coordinates of the Point of Intersection:
The normal chord intersects the parabola at another point. Let's find the coordinates of this point.

Substituting the equation of the normal chord into the equation of the parabola, we have:
(2ahy - 2ahk)^2 = 4ax.

Expanding and simplifying this equation, we get:
4a^2h^2y^2 - 8a^2h^2yk + 4a^2h^2k^2 = 4ax.

Dividing both sides of the equation by 4ah^2, we obtain:
ay^2 - 2ayk + k^2 = x.

Since this equation represents a parabola, the coefficient of y^2 must be equal to 4a. Therefore, we have:
4a = a => a = 1.

Simplifying the equation further, we get:
y^2 - 2yk + k^2 = x.

Comparing this equation with the equation of the parabola, we can conclude that the point of intersection of the normal chord and the parabola is (k, k).

3. Proving that the Normal Chord Subtends a Right Angle:
To prove that the normal chord subtends a right angle at the focus, we need to show that the product of the slopes of the two lines is -1.

Let's consider the line passing through the points (h, k) and (k, k). The slope of this line is (k - k)/(h - k) = 0.

The slope of the line passing through the focus (0, 0) and the point (k, k) is k/k = 1.

The product of these slopes is 0 * 1 = 0.

Since the product of the slopes is 0, which is the negative reciprocal of -1, we can conclude that the normal chord subtends a right angle at the focus.

Hence, the normal chord to the parabola y^2 = 4ax at the point whose ordinate is equal to the abscissa subtends a right angle at the focus.