Statement 1: If in a triangle, orthocentre, circumcentre and centroid are rational points, then its vertices must also be rational points.Statement 2: If the vertices of a triangle are rational points, then the centroid, circumcentre are also rational points.

Statement 2 is obviously correct.For statement 1, let thecircumcentre be at (0,0) and the vertices of the triangle be (x1,y1),(x2,y2) and (x3,y3). Then centroid is (x1+x2+x33,y1+y2+y33), and orthocentre of the triangles becomes (x1+x2+x3,y1+y2+y3).This implies that if the centroid is rational then orthocentre is also rational but (x1+x2+x3) can be rational even if x1,x2,x3 are not all rational.For example, A(0,1),B(−1/2,3/2) and C(−1/2,−3/2) where G, H and C are at (0,0) i.e., rational points