Carlyle circle

In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. Given the quadratic equation x2 − sx + p = 0 the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation. The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is x(x − s) + (y − 1)(y − p) = 0. The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle) x2 − sx + p = 0. The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation z5 − 1 = 0. One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation z4 + z3 + z2 + z + 1 = 0. These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp (2ipi/5). Let these correspond to the points P1, P2, P3, P4. Letting p1 = ω + ω4, p2 = ω2 + ω3 we have p1 + p2 = −1, p1p2 = −1. (These can be quickly shown to be true by direct substitution into the quartic above and noting that ω6 = ω, and ω7 = ω2.) So p1 and p2 are the roots of the quadratic equation x2 + x − 1 = 0. The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct p1 and p2. From the definitions of p1 and p2 it also follows that p1 = 2 cos(2pi/5), p2 = 2 cos(4pi/5). These are then used to construct the points P1, P2, P3, P4.