If the vertices P, Q, R of a triangle PQR are | vedclass

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(B) Let the vertices be $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ where all $x_i, y_i \in \mathbb{Q}$.The centroid is given by $(\frac{x_1+x_2+x_3}{3},

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(B) Let the vertices be $P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3)$ where all $x_i, y_i \in \mathbb{Q}$.The centroid is given by $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$,which is always rational.The incentre is given by $(\frac{ax_1+bx_2+cx_3}{a+b+c}, \frac{ay_1+by_2+cy_3}{a+b+c})$,where $a, b, c$ are side lengths. Since $a, b, c$ involve square roots of rational numbers (e.g.,$c = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$),the incentre is not necessarily rational.Similarly,the circumcenter and orthocenter are also not necessarily rational points for a triangle with rational vertices.

If the vertices $P, Q, R$ of a triangle $PQR$ are rational points (points with rational coordinates),which of the following points is/are not necessarily a rational point?

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