The Droz-Farny circles are a pair of circles of equal radius obtained by particular geometric constructions.
The following amazing property of a triangle, first given by Steiner and then proved by Droz-Farny (1901), is related to these circles. Draw a circle
with center at the orthocenter which cuts the lines , , and (where are the midpoints of their
respective sides) at ,
;
,
;
and ,
respectively, then the line segments are all equal:
Conversely, 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.
There is a beautiful generalization of the Droz-Farny circles motivated by the observation that the orthocenter and circumcenter
are isogonal conjugates. Let and be any pair of isogonal
conjugates of a triangle , and let , , and be the feet of the perpendiculars to the sides from one of
the points (say, ),
and let circles with centers , , and be drawn to pass through . Then the three pairs of points on the sides of which are determined by these circles always lie on
a circle with center , and the two circles constructed in this way are congruent
(Honsberger 1995).