Show that D(-1, 4, -3) is the circumcentre of triangle ABC with vertices A(3, 2, -5), B(-3. 8, -5) and C(-3, 2, 1).

To prove: D is circumcenter of triangle ABC

Let us consider D as circumcenter of triangle ABC.

∴ AD = BC = CD.

Formula:

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

D=

Here,

(x₁,y₁,z₁)= (3, 2, -5)

(x₂,y₂,z₂)= (-3. 8, -5)

(x₃,y₃,z₃)= (-3, 2, 1)

(x₄,y₄,z₄) = (-1, 4, -3)

Length AD =

=

=

=

=

Length BD =

=

=

=

=

Length CD =

=

=

=

=

Hence, the condition is consistent.

Hence, D is circumcenter of triangle ABC.