![[(x - 4)^2 - (x - 1)^2] + [(y - 5)^2 - (y - 2)^2] + [(z - 6)^2 - (z - 3)^2] = 0](https://soloferat.biz.id/images/x---42---x---12--y---52---y---22--z---62---z---32--0.jpg)
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- However, if we reinterpret the question as seeking a fourth point $D = (x, y, z)$ such that all six edges $AB, AC, AD, BC, BD, CD$ are equal and the coordinates are integers, we proceed by solving the system:
- Let $AB = BC = CD = DA = DB = DC = s$. Then we solve the system of equations:
- Subtracting the first equation from the second and the second from the third gives linear equations in $x, y, z$. Subtracting the first from the second:
- -6x -6y -6z + 63 = 0 \Rightarrow x + y + z = 10.5
- This is not possible with integer coordinates. Hence, no such integer-coordinate point $D$ exists.
- Therefore, the answer is:
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