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).
