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.

Question

Infinity Learn · Accepted Answer

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

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.

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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.

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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.