Question

Statement 1: If the vertices of a triangle are having rational coordinates, then its centroid, circumcentre and orthocentre are rational.

Statement 2: In any triangle, orthocentre, centroid and circumcentre are collinear and centroid divides the line joining orthocentre and circumcentre in the ratio $2:1$.

Answer 1

Let $(x_1,y_1),(x_2,y_2)\text{and}(x_3,y_3)$ be the rational coordinates of the vertices of the triangle $ABC$.



Centriod$=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})$

This is a rational number.

Orthocentre is the point of intersection of altitudes which will bear rational coefficients when expressed as a straight line. So, orthocentre is also rational.
Circumcentre is the point of intersection of perpendicular bisectors which will bear rational coefficient when expressed as a straight line. So, circumcentre is also rational.
But statement 2 is false as in an equilateral triangle all the centers coincide.