Find the co ordinates of point P which is equaidsistant from the four ponts A(0,0,0) ,B(1,0,0) C(0,2,0) D(0,0,3)

Answer

To find the coordinates of point P, which is equidistant from the four points A(0,0,0), B(1,0,0), C(0,2,0), and D(0,0,3), we will proceed step by step.

### Step 1: Set up the equations for distances
Let the coordinates of point P be (x, y, z). Since point P is equidistant from points A, B, C, and D, we can write the following equations based on the distance formula:

1. Distance from A to P:
\[ AP^2 = x^2 + y^2 + z^2 \]

2. Distance from B to P:
\[ BP^2 = (x - 1)^2 + y^2 + z^2 \]

3. Distance from C to P:
\[ CP^2 = x^2 + (y - 2)^2 + z^2 \]

4. Distance from D to P:
\[ DP^2 = x^2 + y^2 + (z - 3)^2 \]

Since all distances are equal, we can set these equations equal to each other.

### Step 2: Equate AP² and BP²
Setting \( AP^2 = BP^2 \):
\[ x^2 + y^2 + z^2 = (x - 1)^2 + y^2 + z^2 \]
Cancelling \( y^2 + z^2 \) from both sides:
\[ x^2 = (x - 1)^2 \]
Expanding the right side:
\[ x^2 = x^2 - 2x + 1 \]
Cancelling \( x^2 \) from both sides:
\[ 0 = -2x + 1 \implies 2x = 1 \implies x = \frac{1}{2} \]

### Step 3: Equate AP² and CP²
Now, set \( AP^2 = CP^2 \):
\[ x^2 + y^2 + z^2 = x^2 + (y - 2)^2 + z^2 \]
Cancelling \( x^2 + z^2 \) from both sides:
\[ y^2 = (y - 2)^2 \]
Expanding the right side:
\[ y^2 = y^2 - 4y + 4 \]
Cancelling \( y^2 \) from both sides:
\[ 0 = -4y + 4 \implies 4y = 4 \implies y = 1 \]

### Step 4: Equate AP² and DP²
Now, set \( AP^2 = DP^2 \):
\[ x^2 + y^2 + z^2 = x^2 + y^2 + (z - 3)^2 \]
Cancelling \( x^2 + y^2 \) from both sides:
\[ z^2 = (z - 3)^2 \]
Expanding the right side:
\[ z^2 = z^2 - 6z + 9 \]
Cancelling \( z^2 \) from both sides:
\[ 0 = -6z + 9 \implies 6z = 9 \implies z = \frac{3}{2} \]

### Final Coordinates
Thus, the coordinates of point P are:
\[ P\left(\frac{1}{2}, 1, \frac{3}{2}\right) \]