Statement-I : If (2, 3, 5), (3, 5, 2) and (5, 2, 3) are the vertices of a triangle then its circumcentre is $\left(\frac{10}{3},\frac{10}{3},\frac{10}{3}\right)$

Statement - II : If (2, 3, 5), (3, 5, 2) and (5, 2, 3) are the vertices of a triangle then its perimeter is $3\sqrt{3}$

Which of the above statements is correct :

The given vertices are $A(2, 3, 5), B(3, 5, 2) and C (5, 2, 3)$

The lengths of the sides are 

$$\begin{gathered} a=BC=\sqrt{4+9+1}=\sqrt{14} \\ b=CA=\sqrt{9+1+4}=\sqrt{14} \\ c=AB=\sqrt{1+4+9}=\sqrt{14} \end{gathered}$$

Observing the above sides, the triangle formed by the given vertices is equilateral triangle. 

SInce the triangle is equilateral triangle, the circum center is coincie with centroid

Hence 

$\begin{array}{rcl}S& =& G=\left(\frac{2+3+5}{3},\frac{3+5+2}{3},\frac{5+2+3}{3}\right)\\ & =& \left(\frac{10}{3},\frac{10}{3},\frac{10}{3}\right)\end{array}$

This is true.

The perimeter of the traingle is $3\sqrt{14}$

Hence, the above statement II is false.