IIT JEE 1984

The key observation is that the intersection of two straight lines with equations of the form $a_1 x+b_1 y=c_1$ and $a_2 x+b_2 y=c_2$ , where $a_i,b_i,c_i$ are rational numbers, will always have rational coordinates (as long as the lines are not parallel). Call such a straight line "rational" for convenience below.

Further, the line joining any two rational points is rational, and any line perpendicular to a line joining two rational points will also be rational. (The way to see this is that, algebraically, we're solving simultaneous linear equations with rational coefficients; we never have to take any square roots.)

Using this on the centres...

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Orthocentre: this is the intersection of the three altitudes of the triangle. The altitudes are all rational lines; so they always intersect in a rational point.

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Circumcentre: this is the intersection of the three perpendicular bisectors of the sides of the triangle. These are all rational lines; so again they always intersect in a rational point.

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Centroid: the intersection of the medians; the medians are rational lines, so the centroid is rational.

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Incentre: the odd one out. Note that this is constructed using angle bisectors, which are not necessarily rational lines. A simple counterexample is the triangle with vertices $(0,0)$ , $(1,0)$ and $(0,1)$ .