where for arbitrary real values of , , , , , :

,

,

,

,

,

,

,

,

.

To ease notation, let stand for the line joining and .

Further points can be added. Let , , , .

Amazingly, the lines , and are concurrent; let the point of concurrency be and set .

We can continue this process to get a point corresponding to any integer. In general, the points , and are on the line for all integers and , every integer triplet that sums to zero corresponds to such a line, and the lines , , , … are concurrent at the point .

All nine of the original points lie on a cubic curve; a cubic curve has the property that a non-tangent line through two points passes through a third point of the curve.

In the graphic, a red point indicates a negative number.

You can drag the points 1, 2 or ef, with coordinates , or , respectively.