Steiner gave and Droz-Farny (1901) proved that if equal circles are drawn about the vertices of a triangle (dashed circles
in the above figure), they cut the lines joining the midpoints
of the corresponding sides in six points , , , , , and , which lie on a circle whose
center is the orthocenter. If the radius of these
circles
is the radius of the equal circles
centered on the vertices , , and , and is the radius of the circle
about ,
then

In the special case that , then a circle , known as the Droz-Farny circle is obtained. This circle
has center
and radius whose square is given by

(2)

(3)

(Johnson 1929, pp. 257-278).

Another construction for the first Droz-Farny circle proceeds by drawing circles with centers at the feet of the altitudes and passing through the circumcenter.
These circles cut the corresponding sides in six concyclic points whose circumcircle
is the first Droz-Farny circle.

The first Droz-Farny circle therefore passes through 12 notable points, two on each
of the sides and two on each of the lines joining midpoints of the sides, as illustrated
in the rather busy figure above.