To find the coordinates of the point equidistant from the four points A(0,0,0), B(a,0,0), C(0,b,0), and D(0,0,c), we can follow these steps:
### Step 1: Set up the distance equations
Let the point P be represented by coordinates (x, y, z). We need to find the distances from point P to each of the points A, B, C, and D.
1. Distance from P to A:
\[
PA = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{x^2 + y^2 + z^2}
\]
2. Distance from P to B:
\[
PB = \sqrt{(x - a)^2 + (y - 0)^2 + (z - 0)^2} = \sqrt{(x - a)^2 + y^2 + z^2}
\]
3. Distance from P to C:
\[
PC = \sqrt{(x - 0)^2 + (y - b)^2 + (z - 0)^2} = \sqrt{x^2 + (y - b)^2 + z^2}
\]
4. Distance from P to D:
\[
PD = \sqrt{(x - 0)^2 + (y - 0)^2 + (z - c)^2} = \sqrt{x^2 + y^2 + (z - c)^2}
\]
### Step 2: Set the distances equal
Since point P is equidistant from A, B, C, and D, we can set the distances equal to each other. We will first set \( PA = PB \).
\[
\sqrt{x^2 + y^2 + z^2} = \sqrt{(x - a)^2 + y^2 + z^2}
\]
### Step 3: Square both sides
Squaring both sides to eliminate the square roots gives us:
\[
x^2 + y^2 + z^2 = (x - a)^2 + y^2 + z^2
\]
### Step 4: Simplify the equation
Expanding the right side:
\[
x^2 + y^2 + z^2 = x^2 - 2ax + a^2 + y^2 + z^2
\]
Now, cancel \( x^2, y^2, z^2 \) from both sides:
\[
0 = -2ax + a^2
\]
### Step 5: Solve for x
Rearranging gives us:
\[
2ax = a^2 \implies x = \frac{a}{2}
\]
### Step 6: Repeat for the other points
Now we can repeat the process for \( PA = PC \):
\[
\sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + (y - b)^2 + z^2}
\]
Squaring both sides:
\[
x^2 + y^2 + z^2 = x^2 + (y - b)^2 + z^2
\]
Simplifying gives:
\[
0 = (y - b)^2 \implies y = \frac{b}{2}
\]
And for \( PA = PD \):
\[
\sqrt{x^2 + y^2 + z^2} = \sqrt{x^2 + y^2 + (z - c)^2}
\]
Squaring both sides:
\[
x^2 + y^2 + z^2 = x^2 + y^2 + (z - c)^2
\]
Simplifying gives:
\[
0 = (z - c)^2 \implies z = \frac{c}{2}
\]
### Final coordinates
Thus, the coordinates of the point P that is equidistant from A, B, C, and D are:
\[
P\left(\frac{a}{2}, \frac{b}{2}, \frac{c}{2}\right)
\]
