A circle that in internally tangent to two sides of a triangle and to the circumcircle is called a mixtilinear incircle. There are three mixtilinear incircles, one corresponding to each angle of the triangle.
The radius of the 
-mixtilinear
incircle inscribed in 
is given by
 |
(1)
|
where 
is the inradius of the reference
triangle (Durell and Robson 1935), and the center function is
 |
(2)
|
These can be derived by considering exact trilinear coordinates 
and noting the condition that the 
-circle be tangent to sides 
and 
means that
 |
(3)
|
Let 
be the distance between the centers of the 
-circle and circumcircle which can be found using the trilinear
distance formula, then since these two circles are tangent internally,
 |
(4)
|
Combining this with the condition for exact trilinears 
then gives two equation that can
be solved for the two unknowns 
and 
.
The points of contact of the 
-mixtilinear incircle with the sides 
and 
are found by intersecting these sides with the line through
the incenter 
perpendicular to the angle bisector 
(Veldkamp 1976-1977).
The radical line of the two mixtilinear incircles tangent to side 
passes through the midpoint of the arc 
not containing 
, and through the midpoint of the tangent radius of the incircle
to 
(Nguyen and Salazar 2006).
Let 
be the point of tangency of the mixtilinear incircle inscribed in 
and the circumcircle. Similarly define 
and 
. Then the lines 
, 
, and 
are concurrent. The point of concurrency is the external
similitude center of the circumcircle and incircle, which is Kimberling center 
(Yiu 1999).
The triangle joining the centers is the mixtilinear triangle and its circumcircle is the mixtilinear circle.
