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Prove that the points A(0, 4, 1), B(2, 3...

Prove that the points A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2) are vertices of a square.

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Here, `AB= sqrt((2-0)^(2)+(3-4)^(2)+(-1-1)^(2))=sqrt(4+1+4)=3` units
`BC = sqrt((4-2)^(2)+(5-3)^(2)+(0+1)^(2))=sqrt(4+4+1)=3` units
`CD= sqrt((2-4)^(2)+ (6-5)^(2) + (2-0)^(2))=sqrt(4+1+4)=3` units
`DA = sqrt((0-2)^(2)+(4-6)^(2)+(1-2)^(2))=sqrt(4+4+1)=3` units
`therefore AB = BC= CD=DA`
`therefore` ABCD is a rhombus

Now `AC = sqrt((4-0)^(2)+(5-4)^(2)+(0-1)^(2))=sqrt(16+1+1)=3sqrt(2)` units
`BD = sqrt((2-2)^(2)+(6-3)^(2)+(3+1)^(2))=sqrt(9+9)=3sqrt(2)` units
Now, ABCD is a rhombus of equal diagonals.
Hence, ABCD is a square.
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