Given a point 
and a triangle 
,
the Miquel triangle is the triangle 
connecting the side points 
, 
, and 
of 
with respect to which 
is the Miquel point.
Let the points defining the Miquel circles be fractional distances 
,
, and 
along the sides 
, 
, and 
, respectively, and let 
and 
. The Miquel triangle has side lengths
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
and area
![Delta_M=[1-(k_a+k_b+k_c)+k_ak_b+k_bk_c+k_ck_a]Delta,](/images/equations/MiquelTriangle/NumberedEquation1.svg) |
(4)
|
where 
is the area of the reference
triangle.
In the special case 
,
the Miquel triangle becomes the medial triangle.
All Miquel triangles of a given point 
are directly similar, and 
is the similitude center
in every case.
