Zero-Sum Triplet Curves

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Consider the magic square whose rows, columns and diagonals sum to zero:


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.


Contributed by: Ed Pegg Jr (January 2024)
Open content licensed under CC BY-NC-SA



With mirror symmetry, the initial points are , , , , and . Green lines go through a point at infinity.

With skew symmetry, the initial points are , , , , , and .

You can drag the point mn with coordinates .

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