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Carlyle Circle


CarlyleCircle

Consider a quadratic equation x^2-sx+p=0 where s and p denote signed lengths. The circle which has the points A=(0,1) and B=(s,p) as a diameter is then called the Carlyle circle C_(s,p) of the equation. The center of C_(s,p) is then at the midpoint of AB, M=(s/2,(1+p)/2), which is also the midpoint of S=(s,0) and Y=(0,1+p). Call the points at which C_(s,p) crosses the x-axis H_1=(x_1,0) and H_2=(x_2,0) (with x_1<=x_2). Then

 s=x_1+x_2
(1)
 p=x_1x_2
(2)
 (x-x_1)(x-x_2)=x^2-sx+p,
(3)

so x_1 and x_2 are the roots of the quadratic equation.


See also

257-gon, 65537-gon, Heptadecagon, Pentagon

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References

Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, pp. 4-5, 1982.DeTemple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97-108, 1991.Eves, H. An Introduction to the History of Mathematics, 6th ed. Philadelphia, PA: Saunders, 1990.Leslie, J. Elements of Geometry and Plane Trigonometry with an Appendix and Very Copious Notes and Illustrations, 4th ed., improved and exp. Edinburgh: W. & G. Tait, 1820.

Referenced on Wolfram|Alpha

Carlyle Circle

Cite this as:

Weisstein, Eric W. "Carlyle Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CarlyleCircle.html

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