Carlyle circle
In mathematics, a Carlyle circle (after Thomas Carlyle (1795–1881)) is a certain circle in a coordinate plane associated with a quadratic equation. 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
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
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)
The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation
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
These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp(2πi/5). Let these correspond to the points P1, P2, P3, P4. Letting
we have
So p1 and p2 are the roots of the quadratic equation
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
These are then used to construct the points P1, P2, P3, P4.
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