Geometry and Arithmetic

Everyone knows what a rational number is, a quotient of two integers. We call a point (x, y) in the plane a rational point if both of its coordinates are rational numbers. We call a line a rational line if the equation of the line can be written with rational numbers, that is, if it has an equation $$\displaystyle{ax + by + c = 0}$$ with a, b, and c rational. Now it is pretty obvious that if you have two rational points, then the line through them is a rational line. And it is neither hard to guess nor hard to prove that if you have two rational lines, then the point where they intersect is a rational point. Equivalently, if you have two linear equations with rational numbers as coefficients and you solve them, you get rational numbers as answers.