Find the coordinates of a point equidistant from the four points O(0,0,0), A(p,0,0), B(0,q,0) and C(0,0,r)

Answer

To find the coordinates of a point that is equidistant from the four points O(0,0,0), A(p,0,0), B(0,q,0), and C(0,0,r), we can use the concept of the centroid of the tetrahedron formed by these points.

### Step-by-Step Solution:

1. **Identify the Coordinates of the Points:**
- O = (0, 0, 0)
- A = (p, 0, 0)
- B = (0, q, 0)
- C = (0, 0, r)

2. **Use the Formula for the Centroid:**
The coordinates of the centroid (G) of a tetrahedron formed by the points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3), and (x4, y4, z4) are given by:
\[ G = \left( \frac{x1 + x2 + x3 + x4}{4}, \frac{y1 + y2 + y3 + y4}{4}, \frac{z1 + z2 + z3 + z4}{4} \right) \]

3. **Substitute the Coordinates into the Formula:**
- For the x-coordinate:
\[ G_x = \frac{0 + p + 0 + 0}{4} = \frac{p}{4} \]
- For the y-coordinate:
\[ G_y = \frac{0 + 0 + q + 0}{4} = \frac{q}{4} \]
- For the z-coordinate:
\[ G_z = \frac{0 + 0 + 0 + r}{4} = \frac{r}{4} \]

4. **Combine the Coordinates:**
Therefore, the coordinates of the point G that is equidistant from the points O, A, B, and C are:
\[ G = \left( \frac{p}{4}, \frac{q}{4}, \frac{r}{4} \right) \]

### Final Answer:
The coordinates of the point equidistant from O(0,0,0), A(p,0,0), B(0,q,0), and C(0,0,r) are:
\[ \left( \frac{p}{4}, \frac{q}{4}, \frac{r}{4} \right) \]