Droz-Farny Circles
Draw a Circle with center 
which cuts the lines 
, 
, and 
(where 
are theMidpoints) at 
, 
; 
, 
; and 
, 
respectively, then
Conversely, if equal Circles are drawn about the Vertices of a Triangle,they cut the lines joining the Midpoints of the corresponding sides in six points. These points lie on aCircle whose center is the Orthocenter. If
is the Radius of the equal Circlescentered on the vertices 
, 
, and 
, and 
is the Radius of the Circle about , then
If the circles equal to the Circumcircle are drawn about the Vertices of a triangle, theycut the lines joining midpoints of the adjacent sides in points of a Circle 
with center and Radius
It is equivalent to the circle obtained by drawing circles with centers at the feetof the altitudes and passing through the Circumcenter. These circles cut the corresponding sides in six points on acircle 
whose center is .
Furthermore, the circles about the midpoints of the sides and passing though cut the sides in six points lying on anotherequivalent circle 
whose center is 
. In summary, the second Droz-Farny circle passes through 12 notable points, twoon each of the sides and two on each of the lines joining midpoints of the sides.
References
Goormaghtigh, R. ``Droz-Farny's Theorem.'' Scripta Math. 16, 268-271, 1950. 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.
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