## Carlyle Circle

Consider a Quadratic Equation where and denote signed lengths. The Circle which has the points and as a Diameter is then called the Carlyle circle of the equation. The Center of is then at the Midpoint of , , which is also the Midpoint of and . Call the points at which crosses the x-Axis and (with ). Then

so and are the Roots of the quadratic equation.

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.