If the coordinates of the vertices of a triangle an: rational numbers, then which of the following points of the triangle may not have rational coordinates?

Let $A\left(x_1,y_1\right),B\left(x_2,y_2\right)$ and $C\left(x_3,y_3\right)$

vertices of $\mathrm{△}ABC$ such that $x_1,x_2,x_3,y_1,y_2,y_3$  are rational numbers. Then,

$\frac{x_1+x_2+x_3}{3}$ and $\frac{y_1+y_2+y_3}{3}$ are also rational numbers

So, the coordinates of the centroid are always rational numbers.

Let $P$ be the circumcentre of .$∆ABC$. Then

$\begin{array}{r}PA=PB=PC\\ \Rightarrow P{A}^2=P{B}^2=P{C}^2\end{array}$

$\Rightarrow P{A}^2=P{B}^2\text{ and }P{B}^2=P{C}^2$