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+---- Thread: Carlyle circle [wikipedia] (/showthread.php?tid=458)
Carlyle circle [wikipedia] - mklabgr - 06-17-2026
The Carlyle circle is a geometric construction that connects quadratic equations with circles. For a quadratic equation of the form $x^2−sx+p=0$, one constructs a circle having the segment between the points (0,1) and (s,p) as its diameter. A remarkable property of this circle is that the xxx-coordinates of its intersections with the xxx-axis are exactly the roots of the quadratic equation.
Named after the Scottish thinker and mathematician Thomas Carlyle, the Carlyle circle provides an elegant geometric method for solving quadratic equations and has also been used in compass-and-straightedge constructions of regular polygons such as pentagons and higher-order polygons.
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