For the following two (02) items: Suppose S is the sphere with the...

The points $A(1, 0, 0)$, $B(0, 1, 0)$, and $C(0, 0, 1)$ form an equilateral triangle in 3D space.

The side length of this triangle is $AB = \sqrt{(1-0)^2 + (0-1)^2 + (0-0)^2} = \sqrt{2}$.

The sphere with the smallest radius passing through three given points has its center at the circumcenter of the triangle formed by these points, and its radius is equal to the circumradius of the triangle.

For an equilateral triangle of side length $a$, the circumradius is given by $R = \dfrac{a}{\sqrt{3}}$.

Substituting $a = \sqrt{2}$, we get $R = \dfrac{\sqrt{2}}{\sqrt{3}} = \sqrt{\dfrac{2}{3}}$.

Alternatively, the circumcenter of the equilateral triangle is its centroid $G\left(\dfrac{1}{3}, \dfrac{1}{3}, \dfrac{1}{3}\right)$.

The radius is the distance $AG = \sqrt{\left(1 - \dfrac{1}{3}\right)^2 + \left(0 - \dfrac{1}{3}\right)^2 + \left(0 - \dfrac{1}{3}\right)^2} = \sqrt{\dfrac{4}{9} + \dfrac{1}{9} + \dfrac{1}{9}} = \sqrt{\dfrac{6}{9}} = \sqrt{\dfrac{2}{3}}$.

Answer: $\sqrt{\dfrac{2}{3}}$