Introduction:
We are given the coordinates of points A(2,3,7) and B(-1,3,2) and need to find the coordinates of point C such that the median through point A is equally inclined to the coordinate axes. We will use the concept of medians and slopes to solve this problem.

Method:
1. Let's start by finding the midpoint M of line segment AB.
- The x-coordinate of M is the average of the x-coordinates of A and B.
So, (xM) = (2 + (-1))/2 = 1/2
- Similarly, the y-coordinate of M is the average of the y-coordinates of A and B.
So, (yM) = (3 + 3)/2 = 3
- The z-coordinate of M is the average of the z-coordinates of A and B.
So, (zM) = (7 + 2)/2 = 9/2
Therefore, the coordinates of M are (1/2, 3, 9/2).

2. Now, let's find the slope of the median through A.
- The slope of the median can be found using the formula:
slope = (zC - zA)/(xC - xA) = (r - 7)/(p - 2)
(where C has coordinates (p, 5, r))

3. We are given that the median through A is equally inclined to the coordinate axes.
- For a line to be equally inclined to the coordinate axes, the product of the slopes of the line with each of the coordinate axes should be -1.
- In other words, (slope with x-axis) * (slope with y-axis) * (slope with z-axis) = -1.
- Substituting the slope of the median through A, we get:
[(r - 7)/(p - 2)] * 0 * 1 = -1
(since the slope with the z-axis is 1 and the slope with the y-axis is 0)

4. Simplifying the equation, we get:
(r - 7)/(p - 2) = 0

5. From the above equation, we can conclude that r = 7 and p = 2.

6. Finally, substituting the values of p and r in the coordinates of point C, we get:
C(2, 5, 7)

Conclusion:
The values of p and r are 2 and 7, respectively. The coordinates of point C are (2, 5, 7).