Construction of Lattice Point in Cartesian Plane

Theorem
Let $\CC$ be a Cartesian plane.

Let $P = \tuple {a, b}$ be a lattice point in $\CC$.

Then $P$ is constructible using a compass and straightedge construction.

Proof
Let $O$ denote the point $\tuple {0, 0}$.

Let $A$ denote the point $\tuple {1, 0}$.

The $x$-axis is identified with the straight line through $O$ and $A$.

The $y$-axis is constructed as the line perpendicular to $OA$ through $O$.

From Construction of Integer Multiple of Line Segment, the point $\tuple {a, 0}$ is constructed.

Drawing a circle whose center is at $O$ and whose radius is $OA$ the point $A'$ is constructed on the $y$-axis where $OA' = OA$.

Thus $A'$ is the point $\tuple {0, 1}$.

From Construction of Integer Multiple of Line Segment, the point $\tuple {0, b}$ is constructed.

Using Construction of Parallel Line, a straight line is drawn through $\tuple {a, 0}$ parallel to the $y$-axis.

Using Construction of Parallel Line, a straight line is drawn through $\tuple {0, b}$ parallel to the $x$-axis.

By definition of Cartesian plane, their intersection is at $\tuple {a, b}$, which is the required point $P$.