A mixtilinear incircle is a circle that is internally tangent to the circumcircle of a triangle and tangent to two of its sides. 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 triangle
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,
as found using the distance formula in trilinear
coordinates. Since these two circles are internally
tangent,
 |
(4)
|
Combining this with the condition for exact trilinear coordinates 
then gives two equations
that can be solved for the unknowns 
and 
.
The points of tangency of the 
-mixtilinear incircle with 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
radius of the incircle perpendicular 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.
The two intersections of each mixtilinear incircle with the opposite side of its reference
triangle give six points on a common conic
section (E. A. José García, pers. comm., Oct. 27,
2013). To verify this, write 
, 
, 
, and let 
be the semiperimeter.
The distance from 
to either point of tangency on 
or 
is 
. If 
and 
are the intersections of
the 
-mixtilinear
incircle with 
,
then circle power gives
 |  |  |
(5)
|
 |  |  |
(6)
|
Multiplying the ratio of these expressions and the two cyclically corresponding ratios gives 1, so the conclusion follows from Carnot's conic theorem (Bogomolny).
