Stammler Circles

The Stammler circles are the three circles (apart from the circumcircle), that intercept the sidelines of a reference triangle in chords of lengths equal to the corresponding side lengths , , and . The centers of these circles form an equilateral triangle known as the Stammler triangle.

The radical lines of pairs of the Stammler circles are the Simson lines of the vertices of the circumtangential triangle.

The trilinear coordinates of the center of the -Stammler circle are

(1)

and the squares of the radii , , are given by the roots of the cubic equation

(2)

where is the circumradius of the reference triangle (Ehrmann and van Lamoen 2002).

Explicitly, the radii of the Stammler circles are

			(3)
			(4)
			(5)

where is again the circumradius of the reference triangle (Ehrmann and van Lamoen 2002).

When and are the radii of the Stammler circles and the circumradius, then the following equations hold:

			(6)
			(7)
			(8)

(Ehrmann and van Lamoen 2002).