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
(1)
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where is the circumradius of the reference triangle (Johnson 1929, p. 257).
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)
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(3)
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(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.
The first Droz-Farny circle has circle function
(4)
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No Kimberling centers lie on the first Droz-Farny circle.