Question

If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle PQR is

(are) always rational point(s)

• A. Centroid
• B. Incentre
• C. Circumcentre
• D. Orthocentre

Answer 1

Answer: (A), (C), (D)

The correct options are
A

Centroid

C

Circumcentre

D

Orthocentre

To show that the coordinates of the circumcentre are rational:
Since, he coordinates of midpoint of side PQ is given as D$(\frac{x_1+x_2}{2},\frac{y1+y2}{2})$.
Thus, the equation of perpendicular bisector of PQ is given is
$y-\left(\frac{y_1+y_2}{2}\right)=-\left(\frac{x_1-x_2}{y_1-y_2}\right)(x-(\frac{x_1+x_2}{2}\left)\right)$
Similar way, the equation of perpendicular bisector of QR is given is
$y-\left(\frac{y_2+y_3}{2}\right)=-\left(\frac{x_3-x_2}{y_3-y_2}\right)(x-(\frac{x_2+x_3}{2}\left)\right)$
Similar way, the equation of perpendicular bisector of RP is given is

$y-\left(\frac{y_1+y_3}{2}\right)=-\left(\frac{x_3-x_1}{y_3-y_1}\right)(x-(\frac{x_1+x_3}{2}\left)\right)$
Since $x_1,x_2,x_3,y_1,y_2,y_3$ are rational.
So, any point satisfying all three equations will have rational coordinates.
Now, we know the circumcentre is the point of intersection of the perpendicular bisectors of the sides of the triangle.
Hence, we can conclude that the coordinates of the circumcentre are rational.