More on Newton's Diameter Theorem

2 days ago

Newton's diameter theorem states that for a polynomial curve f(x,y)=0 of degree n, the centroids of intersections with parallel lines lie on a straight line.
The post illustrates the theorem using a fifth-degree polynomial f(x,y)=y³+y-x(x+1)(x+2)(x-3)(x-2) and non-evenly spaced parallel lines y=x+k with k values 0.5, -0.5, and -3, showing centroids at (0.4,0.9), (0.4,-0.1), and (0.4,-2.6).
It emphasizes the necessity for each line to intersect the curve n times by contrasting a line y=x+2 that only crosses three times, shifting the centroid to (0.0550019, 2.055).

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