There are three theorems related to pedal circles that go under the collective title of the Fontené theorems.
The first Fontené theorem lets 
be a triangle and 
an arbitrary point, 
be the medial
triangle of 
,
and 
be the pedal triangle of 
with respect to 
. Denote the intersections of the corresponding sides of 
and 
as 
, 
, and 
(e.g., 
is the intersection of 
and 
, etc.), then the lines 
, 
and 
meet at a point 
common to the circumcircles of 
(which is the nine-point
circle of 
)
and 
(which is the pedal circle of 
with respect to 
).
The second Fontené theorem states that if a point moves on a fixed line through the circumcenter, then its pedal circle passes through a fixed point on the nine-point circle, as illustrated above.
The third Fontené theorem states that the pedal circle of a point 
is tangent to the nine-point circle iff 
and its isogonal
conjugate 
lie on a line through the circumcenter.
(Note that Johnson (1929) erroneously states this theorem with the orthocenter
in place of the circumcenter.) Feuerbach's
theorem is a special case of this theorem.
