TOPICS

First Droz-Farny Circle

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)

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) (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.

The first Droz-Farny circle has circle function

 (4)

No Kimberling centers lie on the first Droz-Farny circle.

Central Circle, Second Droz-Farny Circle

References

Droz-Farny, A. "Notes sur un théorème de Steiner." Mathesis 21, 22-24, 1901.Goormaghtigh, R. "Droz-Farny's Theorem." Scripta Math. 16, 268-271, 1950.Honsberger, R. "The Droz-Farny Circles." §7.4 (ix) in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 69-72, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 256-258, 1929.

Referenced on Wolfram|Alpha

First Droz-Farny Circle

Cite this as:

Weisstein, Eric W. "First Droz-Farny Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstDroz-FarnyCircle.html