Ok. Let's say P is (a,b) Q is (c,d) and R is (e,f)

a,b,c,d,e and f are all rational ie they are fractions

Midpoint PQ

and centroid is

the {rationals} are closed for +, - , x and ÷

ie. adding two fractions, or subtracting one fraction from another, or multiplying two fractions or dividing them will always give another fracftion.

Therefore both the midpoint and the centroid have rational coordinates.

Bob