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Doubling a chord on a circle or arc—once externally and once internally—reveals the radius and centre of the underlying circle without bisection or any dimensional information.
When the Chord Doubling Process for Arcs is swept through all possible chord lengths, it produces two distinct curves.
The first is a perfect ellipse.
The second is the Chord-Doubling P-locus, traced by the centre of the congruent circle. It is a sextic curve with several notable properties, including being rational and tricircular.
The most surprising outcome is that this P-locus is Dürer’s folium—not merely similar to it, but the same curve, scaled by (2R) and rotated 90°.
What makes this especially interesting is that the two curves arise from completely different geometric constructions: Dürer’s folium comes from a rolling-circle construction, while the Chord-Doubling P-locus emerges naturally from the changing chord and congruent-circle construction.
Below are some examples of the Chord Doubling technique applied to various circles and arcs.