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.

© 1996-9

1999-05-26