Carlyle Circle

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 $x_1\geq x_2$). Then

$$s=x_1+x_2$$

$$p=x_1x_2$$

$$(x-x_1)(x-x_2)=x^2-sx+p,$$

so $x_1$ and $x_2$ are the Roots of the quadratic equation.

See also 257-gon, 65537-gon, Heptadecagon, Pentagon

References

De Temple, 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.