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

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To prove: Points A, B, C, D form square. 

Formula: The distance between two points (x1,y1,z1) and (x2,y2,z2) is given by

D = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Here, 

(x1,y1,z1)= (1, 1, 1) 

(x2,y2,z2)= (-2, 4, 1) 

(x3,y3,z3)= (-1, 5, 5) 

(x4,y4,z4)= (2, 2, 5)

Length AB = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Length BC = \(\sqrt{(x_3-x_2)^2+(y_3-y_2)^2+(z_3-z_2)^2}\)

Length AD = \(\sqrt{(x_4-x_1)^2+(y_4-y_1)^2+(z_4-z_1)^2}\)

Here, AB = BC = CD = AD 

Also, AC = BD 

This means all the sides are the same and diagonals are also equal. 

Hence vertices A, B, C, D form a square.

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