Carlyle Circles

Carlyle circles

Thomas Carlyle was a Scottish philosopher and teacher, and discovered a geometric interpretation for solving quadratic equations. If you start with the equation x2 – sx + p = 0, and construct the circle with the line joining (0, 1) and (s, p) as a diameter, then the abscissas of the points where the circle intersects the x-axis are exactly the roots of the polynomial.

For instance, consider x2 – 4x + 3 = 0. Its Carlyle circle intersects the x-axis in 1 and 3, which are indeed the solutions of the quadratic equation!

Carlyle circles can be used for developing explicit ruler-and-compass constructions of regular polygons.