1. Tardigrade
  2. Question
  3. Mathematics
  4. Statement-1: If in a triangle orthocentre, circumcentre and centroid are rational points then its vertices must also be rational points. because Statement-2: If the vertices of a triangle are rational point then the centroid, circumcentre and orthocentre are also rational points.

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Straight Lines

Solution:

$ S -2$ is obviously correct.
For statement-1: Let the circumcentre be at $(0,0)$ and the vertices of the triangle be $\left(x_1, y_1\right),\left(x_2, y_2\right)$ and $\left(x_3, y_3\right)$ then centroid is $\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}$ and orthocenre of the triangles becomes $\left( x _1+ x _2+ x _3, y _1+ y _2+ y _3\right)$.
This implies that if the centroid is rational then orthocentre is also rational but $\left(x_1+x_2+x_3\right)$ can be rational even if $x _1, x _2, x _3$ are not all rational.
e.g. $A (1,0) ; B (-1 / 2, \sqrt{3} / 2) ; C (-1 / 2,-\sqrt{3} / 2)$ where $G / H / C \cdot C$ are at $(0,0)$ i.e. rational points