Let the points defining the Miquel circles be fractional distances , ,
and along the sides , ,
and , respectively, and let . Then the Miquel point has trilinear coordinates
, where
(1)
(2)
(3)
In the special case ,
the Miquel point becomes the circumcenter.
If and are inscribed in a reference
triangle
and also in the same circle, then their Miquel points and are isogonal conjugates. The angle that , and make to the respective sides of and the angle that , and make to these sides are supplementary.
The pedal triangle is a special case.
, 
,
and 
along the sides 
, 
,
and 
, respectively, and let 
. Then the Miquel point has trilinear coordinates
, where
,
the Miquel point becomes the circumcenter.
and 
are inscribed in a reference
triangle 
and also in the same circle, then their Miquel points 
and 
are isogonal conjugates. The angle that 
, 
and 
make to the respective sides of 
and the angle that 
, 
and 
make to these sides are supplementary.
The pedal triangle is a special case.
