I want to give my students the coordinates of three points and ask them to find the coordinates of the orthocenter. This is a fairly long problem that can involve finding the equations of the altitudes and then finding their intersection.
A long problem needs to run fairly smoothly, because if we hit a bump we might not sort it all out before the dismissal bell. And then, I think, the time was not very productive.
I can pick the original six coordinates so that the equations of all three sides of the triangle have intercepts that are integers. But so far I have not figured out how to make the coordinates of the orthocenter integers.
Recently I asked such a question here, and some helpful person responded that anyone teaching my subject ought to be able to answer such a question himself. So I would remind this helpful person that I have not despaired of finding an answer myself. But life is short, and I hope that someone out there (probably a real math teacher) has already done the work.